Analysis of Algorithms - Elementary graphs algorithms -
|
|
- Myron Griffin
- 6 years ago
- Views:
Transcription
1 Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsach Cntr) anras.rmahl@mh.s Autumn 00 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring Oftn us to rprsnt iffrnt kins of rlational information Dpnncis btwn ntitis Distanc btwn ntitis Graphs In many applications it is intrsting to sarch or travrs th graph Th last numbr of subway stations to gt from Hässlby to Norsborg? How o I visit all subway stations in an fficint mannr? This an following lcturs will al with algorithms for this, an som applications Rprsnting graphs Thr ar many ways of rprsnting graphs A graph G = (V, E) consists of vrtics (nos) V an gs E 5 4 V = {,,,4,5} E = {(,),(,5),(,),(,5),(,4),(4,5)} Th rprsntations iffr in fficincy pning on th algorithm, whthr th graph is ns or spars, an othr factors W will consir two rprsntations: Ajacncy-list rprsntation Ajacncy-matrix rprsntation Rprsnting graphs Rprsnting graphs Unirct graph 5 4 Dirct graph Ajacncy-list rprsntation Ajacncy-matrix rprsntation Ajacncy-list rprsntation Ajacncy-matrix rprsntation
2 Ajacncy-list rprsntation Ajacncy-list rprsntation G = (V,E) rprsnt by an array Aj of V lists, on pr vrtx Aj [u] contains pointrs to (ID s of) all vrtics ajacnt to u Not ajacnt is iffrnt whn th graph is irct or unirct Dirct: sum of lngth of all ajacncy lists is E Unirct: sum of lngth of all ajacncy lists is E Mmory rquir: Θ(V + E) 4 Oftn fficint rprsntation 5 if G is spars 5 E not too big compar with V Easily aapt to wight graphs Stor wight w on g in ajacncy list Ajacncy-list rprsntation Ajacncy-matrix rprsntation Ajacncy-matrix rprsntation: G = (V,E) rprsnt by a boolan matrix A, whr: Siz O( V ) (inpnnt of E ), fficint whn E = Θ( V ) (an graph siz is small) 4 5 Ajacncy matrix symmtric whn graph is unirct, thn suffics to stor on half 0 (still siz O( V ), though) Unwight graph can b stor using on bit pr ntry Easily aapt to wight graphs Stor wight w for g (u,v) at ntry in row u an column v Ajacncy-matrix rprsntation Travrsing graphs Most basic algorithms on graphs will b applications of graph travrsal. Printing or valiating ach g/vrtx. Copying a graph or convrting btwn rprsntations. Counting th numbr of gs/vrtics. Intifying connct componnts. Fining paths btwn two vrtics, or cycls. Travrsing graphs Efficincy an corrctnss Efficincy: Don t loop or visit vrtics rpatly. Corrctnss: Don t miss any vrtx. W n to mark vrtics as w travrs th graph. Uniscovr (whit), th initial stat, bfor w v sn it. Discovr (gray), w v sn th vrtx but not all of its incint gs. Finish (black), all incint gs hav bn visit. Orr of xploration Th orr in which w xplor vrtics pns on th containr us for storing iscovr but not finish vrtics. Thr ar two typs of containrs us: Quu: las to so call brath-first sarch. Stack: las to so call pth-first sarch. W will invstigat ths two graph travrsal algorithms in mor tail Brath-first sarch Simpl algorithm for sarching a graph Input: A irct/unirct graph with sourc s Output: Th shortst istanc from s. [u] = shortst istanc of no u to s p[u] = prcssor of no u in shortst path to s Rprsnt by a brath-first tr Only visits vrtics rachabl from s. Running tim: Θ(V+E) Linar tim with rspct to ajacncy list.
3 Brath-first sarch Th BFS algorithm Givn graph G an sourc vrtx s, fin all vrtics rachabl from s by a brath-first sarch from s (whr s is sn as th root in a tr spanning G) Brath-first mans all vrtics at pth k from s ar visit bfor thos at pth k + ar Th algorithm yils th following rsults: Th istanc (lngth of shortst path) from s to any othr rachabl vrtx A s-root brath-first tr that consists of th shortst paths from s to all othr vrtics Works on both irct an unirct graphs Each vrtx u has attributs: color[u] - color uring th sarch [u]- istanc from root vrtx p[u] - prcssor in th tr bing built Exampl: Brath-first sarch Aftr start no s has bn a to Q 0 Q s 0 Exampl: Brath-first sarch Aftr s has bn procss an its nighbours r an w hav bn a 0 Q w r Exampl: Brath-first sarch Aftr no w has bn procss an its nighbours t an x hav bn a 0 Q r t x Exampl: Brath-first sarch Aftr no r has bn procss an its nighbour v has bn a 0 Q t x v
4 Exampl: Brath-first sarch Aftr no t has bn procss an its nighbour u has bn a 0 Q x v u Exampl: Brath-first sarch Aftr no x has bn procss an its nighbour y has bn a 0 Q v u y Exampl: Brath-first sarch Aftr no v has bn procss r s t u 0 Q u y v w x y Exampl: Brath-first sarch Aftr no u has bn procss r s t u 0 Q y v w x y Exampl: Brath-first sarch Aftr no y has bn procss r s t u 0 Q «v w x y Analysis of BFS What about th outr whil-loop? First loop taks O( V ) tim What about th innr for-loop? 4
5 Analysis of BFS Outr loop (whil): Can w boun th tim until Q bcoms mpty? Only whit vrtics ar nquu, an thy ar always gray whn nquu ï Thus a vrtx can b nquu at most onc On vrtx quu for vry lap ï Thus outr whil loop xcut O( V ) tims Innr loop (for): Analysis of BFS Th ajacncy list of a vrtx is scann at most onc, an ach lmnt in th list is accss only onc Thus, th total tim in innr loop is proportional to sum of lngths of all ajacncy lists ïo( E ) Whol loop nst ï O( V ) + O( E ) = O( V + E ) Analysis of BFS Analysis of BFS First loop runs in O( V ) tim Scon loop nst runs in O( V + E ) tim ïth total running tim for BFS is O( V ) + O( V + E ) = O( V + E ) Dpth-first sarch Simpl algorithm which sarchs pr in th graph whnvr possibl Input: A irct/unirct graph. Output: Dp-first forst (compos of pth-first trs) Each vrtx u is tim-stamp: iscovr [u] an finish f[u]. Egs { tr, back, forwar or cross g }. Visits all vrtics. Running tim: Θ(V+E) Linar tim with rspct to ajacncy list. Dpth-first sarch Sarchs graph by rcursivly xploring th vrtics in th ajacncy list All vrtics rachabl from a vrtx in th ajacncy list ar rcursivly sarch bfor nxt vrtx in th list is xplor DFS constructs a pth- first forst that contains all vrtics in th graph BFS only buils a tr of vrtics rachabl from som givn root vrtx Th DFS algorithm [u] is timstamp whn vrtx u first iscovr f[u] is timstamp whn sarch of u s ajacncy list is complt All vrtics rachabl from start no visit using rcursiv call All vrtics ar visit onc 5
6 F T T Dpth-first sarch: vrtx classification Tim-stamp vrtics whn thy ar iscovr/finish: [u]: whn iscovr Th vrtx colors ar quivalnt to th following cass: Whit:th initial stat, bfor w v sn it Gray: w v sn th vrtx but not all of its incint gs. f[u]: whn finish Black: all incint gs hav bn visit T B Dpth-first sarch: g classification Egs ar classifi accoring to th following four cass: Tr-g (T) Back-g (B) [u] < [v]: Forwar-g (F) [u] > [v]: Cross-g (C) Cross-gs (C) ar all othr nontr gs C Tr-gs (T) ar gs in th pthfirst forst Back-gs (B) ar non-tr gs conncting a vrtx u to an ancstor v in a pth-first tr Forwar-gs (F) ar non-tr gs (u,v) conncting a vrtx u to a scnant v in a pth-first tr Dpth-first sarch xampl Dpth-first sarch xampl S: { } S: {a} Dpth-first sarch xampl Dpth-first sarch xampl S: {a, } S: {a,, f}
7 Dpth-first sarch xampl Dpth-first sarch xampl S: {a, } S: {a,, g} Dpth-first sarch xampl Dpth-first sarch xampl S: {a, } S: {a} Dpth-first sarch xampl Dpth-first sarch xampl S: {a, b} S: {a, b, } 7
8 Dpth-first sarch xampl Dpth-first sarch xampl S: {a, b} S: {a} Dpth-first sarch xampl Dpth-first sarch xampl S: { } S: {c} Dpth-first sarch xampl Dpth-first sarch xampl S: {c, h} S: {c} 8
9 Dpth-first sarch xampl First loop taks O( V ) tim How many tims will DFS-VISIT b call? How many tims will this loop run? Rcursiv call S: { } Rsult of DFS run is a forst! Analysis of DFS Running tim of DFS: First loop in DFS taks Θ( V ) tim (ach vrtx visit xactly onc) In scon loop with th rcursiv calls, obsrv that DFS visit will b call xactly onc on ach vrtx (rquirs a proof, rally...) For ach vrtx v whr DFS-VISIT(v) is call, th loop in DFS-VISIT is call Aj[v] tims Sinc vv Aj[v] = Θ( E ) th total cost of scon loop in DFS will b Θ( E ) Thus, running tim of DFS is Θ( V + E ) Topological sort A linar orr of all nos in th graph G such that if G contains an g (u,v) thn u appars bfor v in th orring Topological sort is only possibl if th graph is acyclic On application: DAG rprsnts prcnc rlations btwn tasks or vnts Eg btwn tasks if first task must b prform bfor scon task Thn a topological sort givs a possibl schul of th tasks on a singl rsourc On prcnc graph might hav svral possibl schuls Topological sort: algorithm W can us DFS to gt a topological sort Informal scription: Call DFS(G) to comput finishing tims f[u] for all vrtics u Put nos into list so thy ar stor in crasing orr w.r.t. finishing tim A irct way is to first us DFS(G) an thn sort w.r.t. finishing tim f[u] This yils tim Θ( V + E )+ O( V lg V ) = O( V lg V + E ) But asy to moify DFS to comput th sort list on th fly : Just insrt ach vrtx into list immiatly whn finish As no asymptotic complxity to DFS: still Θ( V + E ) / /5 /7 Exampl: Topological sort unrshorts pants blt shirt ti jackt /8 /5 /4 socks shos 7/8 /4 Prcnc graph for how to gt rss watch socks shos watch unrshorts pants blt shirt ti jackt 7/8 / /5 /4 9/0 /8 /7 /5 /4 9/0 DFS algorithm givs th [u] an f[u] inicat Evnts sort accoring to thir finishing tim = topological sort 9
10 Strongly connct componnts A strongly- connct componnt (SCC) of irct graph G = (V,E) is a maximal subst of vrtics C Œ V such that for vry pair of vrtics u an v in C both vrtics ar rachabl from ach othr E A C B Dirct graph D Strongly connct componnts: {A,B,C,E} an {D} Graph Transpos Th transpos of a irct graph G=(V,E) is th graph G T = (V,E T ) such that G T ={(v,u) V x V : (u,v) E } Thus, G T is G with all gs rvrs Both graphs contains th sam nos Both graphs has th sam SCCs G T can b crat in O(V + E) tim 4 5 Original graph G 4 5 Transpos graph G T Fining SCCs W can us algorithms for DFS an graph transposal for fining SCCs of graph G Stps:. Call DFS(G) to comput finishing tims f[u] for ach vrtx u in G. Comput transpos graph G T. Call DFS(G T ), but in h main loop, consir th vrtics in crasing f[u] (as comput in lin ) 4. Each pth-first tr foun in G T forms a SCC in G Original graph G Fining SCCs Fining SCCs. Call DFS(G) to comput finishing tim f[u] for ach vrtx u /4 / /5 /4 /0 8/9 Sort nos on thir finishing tim: ï b,,a,c,,g,f,h /7 5/ Fining SCCs.. Comput G T from original graph 0
11 Fining SCCs. Call DFS(G T ), but in h main loop, consir th vrtics in crasing f[u] (as comput in lin ï b,,a,c,,g,f,h) Dp- first- trs foun: {b,a,}, {c,}, {g,f}, {h} Fining SCCs 4. Each pth- first tr foun in G T forms a SCC in G: {b,a,}, {c,}, {g,f}, {h} 5. Collaps all SCC to on singl no. Th rsult is th acyclic G SCC graph Each vrtx is on SCC ab fg c h Th visits of nos in th scon DFS corrspons to visiting th vrtics in G SCC in topological sort orr Analysis of SCC algorithm. Call DFS(G) to comput finishing tims f[u] for ach vrtx u in G ï Θ(V + E). Comput transpos graph G T ï Θ(V + E). Call DFS(G T ), but in h main loop, consir th vrtics in crasing f[u] (as comput in lin ) ï Θ(V + E) 4. Collaps ach pth-first tr foun in G T to a SCC ï Θ(V + E) Ovrall total tim: Θ(V + E)
Analysis of Algorithms - Elementary graphs algorithms -
Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsarch Cntr) anras.rmahl@mh.s Autumn 004 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationExamples and applications on SSSP and MST
Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous
More informationOutlines: Graphs Part-4. Applications of Depth-First Search. Directed Acyclic Graph (DAG) Generic scheduling problem.
Outlins: Graps Part-4 Applications o DFS Elmntary Grap Aloritms Topoloical Sort o Dirctd Acyclic Grap Stronly Connctd Componnts PART-4 1 2 Applications o Dpt-First Sarc Topoloical Sort: Usin dpt-irst sarc
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn
More informationThe second condition says that a node α of the tree has exactly n children if the arity of its label is n.
CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More information10. EXTENDING TRACTABILITY
Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What
More informationChapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NP-Completeness. Vertex Cover
Coping With NP-Compltnss Chaptr 0 Extning th Limits o Tractability Q. Suppos I n to solv an NP-complt problm. What shoul I o? A. Thory says you'r unlikly to in poly-tim algorithm. Must sacriic on o thr
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationGraphs and Graph Searches
Graphs an Graph Sarhs CS 320, Fall 2017 Dr. Gri Gorg, Instrutor gorg@olostat.u 320 Graphs&GraphSarhs 1 Stuy Ais Gnral graph nots: Col s Basi Graph Nots.pf (Progrss pag) Dpth first gui: Dpth First Sarh
More informationCS 491 G Combinatorial Optimization
CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl
More informationCSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018
CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationHomework #3. 1 x. dx. It therefore follows that a sum of the
Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationRoadmap. XML Indexing. DataGuide example. DataGuides. Strong DataGuides. Multiple DataGuides for same data. CPS Topics in Database Systems
Roadmap XML Indxing CPS 296.1 Topics in Databas Systms Indx fabric Coopr t al. A Fast Indx for Smistructurd Data. VLDB, 2001 DataGuid Goldman and Widom. DataGuids: Enabling Qury Formulation and Optimization
More informationAssociation (Part II)
Association (Part II) nanopoulos@ismll.d Outlin Improving Apriori (FP Growth, ECLAT) Qustioning confidnc masur Qustioning support masur 2 1 FP growth Algorithm Us a comprssd rprsntation of th dtb databas
More informationMinimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More informationCS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12
Aministrivi: CS61B Ltur #33 Autogrr will run this vning. Toy s Rings: Grph Struturs: DSIJ, Chptr 12 Lst moifi: W Nov 8 00:39:28 2017 CS61B: Ltur #33 1 Why Grphs? For xprssing non-hirrhilly rlt itms Exmpls:
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More informationCSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systms Prof. Mark Fowlr ot St #21 D-T Signals: Rlation btwn DFT, DTFT, & CTFT 1/16 W can us th DFT to implmnt numrical FT procssing This nabls us to numrically analyz a signal to find
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationSPH4U Electric Charges and Electric Fields Mr. LoRusso
SPH4U lctric Chargs an lctric Fils Mr. LoRusso lctricity is th flow of lctric charg. Th Grks first obsrv lctrical forcs whn arly scintists rubb ambr with fur. Th notic thy coul attract small bits of straw
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationKernels. ffl A kernel K is a function of two objects, for example, two sentence/tree pairs (x1; y1) and (x2; y2)
Krnls krnl K is a function of two ojcts, for xampl, two sntnc/tr pairs (x1; y1) an (x2; y2) K((x1; y1); (x2; y2)) Intuition: K((x1; y1); (x2; y2)) is a masur of th similarity (x1; y1) twn (x2; y2) an ormally:
More informationThickness and Colorability of Geometric Graphs
Thicknss an Colorability o Gomtric Graphs Stphan Durochr 1 Dpartmnt o Computr Scinc, Univrsity o Manitoba, Winnipg, Canaa Elln Gthnr Dpartmnt o Computr Scinc, Univrsity o Colorao Dnvr, Colorao, USA Dbajyoti
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationPHYS ,Fall 05, Term Exam #1, Oct., 12, 2005
PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What
More informationMinimum Spanning Trees
Minimum Spnning Trs Minimum Spnning Trs Problm A town hs st of houss nd st of rods A rod conncts nd only houss A rod conncting houss u nd v hs rpir cost w(u, v) Gol: Rpir nough (nd no mor) rods such tht:
More informationAn undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V
Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationFrom Elimination to Belief Propagation
School of omputr Scinc Th lif Propagation (Sum-Product lgorithm Probabilistic Graphical Modls (10-708 Lctur 5, Sp 31, 2007 Rcptor Kinas Rcptor Kinas Kinas X 5 ric Xing Gn G T X 6 X 7 Gn H X 8 Rading: J-hap
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More informationSuperposition. Thinning
Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,
More informationShortest Paths in Graphs. Paths in graphs. Shortest paths CS 445. Alon Efrat Slides courtesy of Erik Demaine and Carola Wenk
S 445 Shortst Paths n Graphs lon frat Sls courtsy of rk man an arola Wnk Paths n raphs onsr a raph G = (V, ) wth -wht functon w : R. Th wht of path p = v v v k s fn to xampl: k = w ( p) = w( v, v + ).
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationA RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES
A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES ADRIAAN DANIËL FOKKER (1887-197) A translation of: Ein invariantr Variationssatz für i Bwgung mhrrr lctrischr Massntilshn Z. Phys. 58, 386-393
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationFinal Exam Solutions
CS 2 Advancd Data Structurs and Algorithms Final Exam Solutions Jonathan Turnr /8/20. (0 points) Suppos that r is a root of som tr in a Fionacci hap. Assum that just for a dltmin opration, r has no childrn
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More information64. A Conic Section from Five Elements.
. onic Sction from Fiv Elmnts. To raw a conic sction of which fiv lmnts - points an tangnts - ar known. W consir th thr cass:. Fiv points ar known.. Four points an a tangnt lin ar known.. Thr points an
More informationChapter 6 Folding. Folding
Chaptr 6 Folding Wintr 1 Mokhtar Abolaz Folding Th folding transformation is usd to systmatically dtrmin th control circuits in DSP architctur whr multipl algorithm oprations ar tim-multiplxd to a singl
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationCSC Design and Analysis of Algorithms. Example: Change-Making Problem
CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationThe Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function
A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n
More informationChapter 8: Electron Configurations and Periodicity
Elctron Spin & th Pauli Exclusion Principl Chaptr 8: Elctron Configurations and Priodicity 3 quantum numbrs (n, l, ml) dfin th nrgy, siz, shap, and spatial orintation of ach atomic orbital. To xplain how
More informationCS 6353 Compiler Construction, Homework #1. 1. Write regular expressions for the following informally described languages:
CS 6353 Compilr Construction, Homwork #1 1. Writ rgular xprssions for th following informally dscribd languags: a. All strings of 0 s and 1 s with th substring 01*1. Answr: (0 1)*01*1(0 1)* b. All strings
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationCS553 Lecture Register Allocation I 3
Low-Lvl Issus Last ltur Intrproural analysis Toay Start low-lvl issus Rgistr alloation Latr Mor rgistr alloation Instrution shuling CS553 Ltur Rgistr Alloation I 2 Rgistr Alloation Prolm Assign an unoun
More informationSIGNIFICANCE OF SMITH CHART IN ANTENNA TECHNOLOGY
SIGNIFICANCE OF SMITH CHART IN ANTENNA TECHNOLOGY P. Poornima¹, Santosh Kumar Jha² 1 Associat Profssor, 2 Profssor, ECE Dpt., Sphoorthy Enginring Collg Tlangana, Hyraba (Inia) ABSTRACT This papr prsnts
More informationCPS 616 W2017 MIDTERM SOLUTIONS 1
CPS 616 W2017 MIDTERM SOLUTIONS 1 PART 1 20 MARKS - MULTIPLE CHOICE Instructions Plas ntr your answrs on t bubbl st wit your nam unlss you ar writin tis xam at t Tst Cntr, in wic cas you sould just circl
More informationCS 241 Analysis of Algorithms
CS 241 Anlysis o Algorithms Prossor Eri Aron Ltur T Th 9:00m Ltur Mting Lotion: OLB 205 Businss HW6 u lry HW7 out tr Thnksgiving Ring: Ch. 22.1-22.3 1 Grphs (S S. B.4) Grphs ommonly rprsnt onntions mong
More informationPreview. Graph. Graph. Graph. Graph Representation. Graph Representation 12/3/2018. Graph Graph Representation Graph Search Algorithms
/3/0 Prvw Grph Grph Rprsntton Grph Srch Algorthms Brdth Frst Srch Corrctnss of BFS Dpth Frst Srch Mnmum Spnnng Tr Kruskl s lgorthm Grph Drctd grph (or dgrph) G = (V, E) V: St of vrt (nod) E: St of dgs
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationMCB137: Physical Biology of the Cell Spring 2017 Homework 6: Ligand binding and the MWC model of allostery (Due 3/23/17)
MCB37: Physical Biology of th Cll Spring 207 Homwork 6: Ligand binding and th MWC modl of allostry (Du 3/23/7) Hrnan G. Garcia March 2, 207 Simpl rprssion In class, w drivd a mathmatical modl of how simpl
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationCSI35 Chapter 11 Review
1. Which of th grphs r trs? c f c g f c x y f z p q r 1 1. Which of th grphs r trs? c f c g f c x y f z p q r . Answr th qustions out th following tr 1) Which vrtx is th root of c th tr? ) wht is th hight
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationChemistry 342 Spring, The Hydrogen Atom.
Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x
More informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
More informationMA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.
MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationIndexed Search Tree (Trie)
Indxd Sarch Tr (Tri) Fawzi Emad Chau-Wn Tsng Dpartmnt of Computr Scinc Univrsity of Maryand, Cog Park Indxd Sarch Tr (Tri) Spcia cas of tr Appicab whn Ky C can b dcomposd into a squnc of subkys C 1, C
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationRandom Access Techniques: ALOHA (cont.)
Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationApplied Statistics II - Categorical Data Analysis Data analysis using Genstat - Exercise 2 Logistic regression
Applid Statistics II - Catgorical Data Analysis Data analysis using Gnstat - Exrcis 2 Logistic rgrssion Analysis 2. Logistic rgrssion for a 2 x k tabl. Th tabl blow shows th numbr of aphids aliv and dad
More information