COMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS
|
|
- Antony Reynolds
- 5 years ago
- Views:
Transcription
1 OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson in [] in whih thy show tht th prolm trmining xistn o suh tilings ws shown to NP-omplt.. Introution or this prolm w onsir th pln s gri o unit squrs, n w in igurs to susts o ths squrs. W onsir th omplxity o ounting how mny wys igur n til y l rtngl n n m rtngl whr l n m r oth t lst 2. In th s tht l = m = 2 this prolm n ormult in trms o ounting prt mthings o plnr grphs whih Kstlyn show oul on in polynomil tim [3]. Thus w look t th s ithr l or m is t lst 3; in prtiulr w will show: Thorm.. ounting th numr o wys to til plnr igur with l n m rtngls is #P-omplt i t lst on o l n m is t lst 3 n th othr is t lst 2. Our proo is moiition o tht givn in [] tht ws us to show tht th ssoit ision prolm is NP-omplt. W n ssoit iprtit grph to onjuntion o ooln luss y ltting th luss n vrils nos n hving gs onnting luss to th vrils thy ontin. Th proo in [] rus rom plnr 3-N St y onvrting th ssoit plnr grph into igur whih is till i n only i th plnr 3-N xprssion is stisil. W will o similr rution rom plnr-x3monost. inition.2. Plnr -x3monost ooln xprssion is -x3mono i it th onjuntion o sris o luss suh tht h lus ontins xtly 3 non-ngt vrils n th lus is tru i xtly on o thos vrils is tru. Suh n xprssion is plnr i th ssoit grph is plnr. inlly, plnr -x3monost is th qustion: Givn plnr -x3mono xprssion, is thr n ssignmnt o ooln vlus to h vril suh tht ll luss in th xprssion r tru? ounting solutions o plnr -x3monost xprssions ws shown to #Pomplt y Hunt, Mrth, Rhkrishnn, n Strns in [2]. 2. Proo o th s l = 2, m = 3 W will irst prov our lim or th s l = 2,m = 3, n thn xtn ths rsults to l 2, m 3. In orr to o this w introu vril igurs, lus
2 2 KYL MYR xmpl o Nottion or Tiling o igur igur xmpl 2 o Nottion or Tiling o igur igur 2 igurs, n g igurs tht orrspon to th vril nos, lus nos, n gs rsptivly o th plnr grph ssoit to plnr -x3mono xprssion. ollowing th onvntion o [], w will pit our igurs y th lttrs -, whr th lttr w us is trmin y th position o th squr mo 2 horizontlly n mo 3 vrtilly, n whr th lttr is pitliz i it is ovr y th vrtil r n lowr-s i it is ovr y th horizontl r(s igurs n 2). Th irst igur w onstut is wir. wir will onnt vril igur to lus igur n will trnsmit th vril vlus rom th vril igur to th lus igur. Wirs n til in two wys orrsponing to tru n ls. Th tru tiling lvs squr on h n until, whil th ls tiling ovrs ll th squrs. W hv two typs o wirs; thos tht strts on n thos tht strt on, n h o ths will n on th othr lttr, n mor spiilly w n xtn th wir so tht it ns on ny o tht lttr, w show wir o th irst typ with oth tru n ls tilings in igurs 3 n 4. Two wirs, on o h typ, run long si h othr orm n g igur. W nxt onstrut th vril igur. This igur will onnt to on g igur or h lus in whih th orrsponing vril pprs, thus th siz vris pning on how mny luss vril is in. Thr r only two wys to til this igur: on wy will or th onnting g igurs to hv tru tiling n th othr will or thm to hv ls tiling. Ths two wys orrspon to
3 OUNTING TILINGS Y TWO RS 3 Wir with Tru Tiling igur 3 Wir with ls Tiling igur 4 Vril igur with Tru Tiling igur 5 Vril igur with ls Tiling igur 6 vril ing tru or ls n r shown in igurs 5 n 6 or th s o 3 onnting g igurs. Our lst igur, shown in igur 7, is th lus igur. It tks in thr input g igurs n is till i n only i xtly on o thm hs tru vlu. This is th stp in whih our rution is not prsimonious, tht is, it os not prsrv th numr o solutions. Spiilly, thr r rossing o wirs in this igur suh tht i oth wirs hv ls vlus, thn th rossing is till in two wys, ithr y thr horizontl rs or two vrtil ons. ut w hv onstrut this igur in suh wy tht wirs orrsponing to h st o two o th thr vrils hs thr rossings. Thus h lus igur n til 2 3 = 8 wys i it n til. W hv thus onstrut th thr igurs tht w n, n w n onnt thm ll us o th plnrity o th xprssion. Thus w hv ru th ounting prolm or plnr -x3monost to plnr tiling prolm with rs o lngth 2 n 3 in whih th numr o tilings is 8 tims th numr o solutions or th -x3monost xprssion, whr is th numr o luss. Thus th prolm is #P-omplt.
4 4 KYL MYR lus igur igur 7 m 2 l igur 8 xtnsion rom l = 2,m = 3 to ny l 2, m 3
5 OUNTING TILINGS Y TWO RS 5 3. xtnsion or l 2, m 3 W now xtn ths rsults to longr rs; horizontl r o lngth l 2 n vrtil r o lngth m 3. W o this rpling h lttr - y rtngl s shown in igur 7. It is lr tht vry tiling o th originl igur will giv ris to tiling o th xpn igur, ut or gnrl igurs this xpnsion is not prsimonious. W lim tht this xpnsion is prsimonious or th igurs tht w r onstruting. To show this w just n to hk tht th tilings or h omponnt igur is prsimonious. W strt with th vril omponnt. To s tht this is prsimonious, irst onsir th ottom lt. This is singl squr, n thus it n ovr ithr y horizontl or vrtil r. On tht is hosn, thn w n work roun th igur n s tht thr is only on wy to til th rst o th igur. Similrly with wirs, on w hv ithr tru or ls vlu or th wir givn y th vril igur w work long th wir n thr is only on wy to til th wir. Th lst igur is th lus igur, ut th lus igur is only ompos o wirs with 3 irnt typs o smll igurs, n it is sy to s tht h o thm hv prsimonious xpnsion, n thus th whol igur os. This omplt th proo. 4. utur irtions On possil irtion or utur rsrh is to rstrit this prolm to simply onnt rgions. s mntion ov, in th gnrl s trmining th xistn o tiling is NP-omplt, ut in th simply-onnt s Knyon n Knyon show in [4], tht trmining th xistn o tiling n on in linr tim. Thus thr is unmntl irn twn ths two ss. itionlly, Pk n Yng show in [5] tht thr is lrg st o rtngls or whih tiling simply onnt rgions is NP-omplt n #P-omplt, so it woul intrsting to s i it is possil to ru th siz o tht st to 2. itionlly it woul intrsting to xtn ths rsults to mor gnrl pirs o rtngls in oth th simply n non-simply onnt ss. 5. knowlgmnts I woul lik to thnk J Yng n Kvin ilks or thir vi n guin in writing this ppr. I woul lso lik to thnk Univrsity o Minnsot, Minnpolis or hosting this RU. itionlly I thnk th RTG grnt NS/MS or proviing uning or this rsrh. Rrns. nil uquir, Muri Nivt, ri Rmil, n Mik Roson, Tiling igurs o th pln with two rs., omput. Gom. 5 (995), Hrry. Hunt III, Mhv V. Mrth, Vnktsh Rhkrishnn, n Rihr win Strns, Th omplxity o plnr ounting prolms, orr s./ (998). 3. Pitr W Kstlyn, imr sttistis n phs trnsitions, Journl o Mthmtil Physis 4 (963), no. 2, lir Knyon n Rihr Knyon, Tiling polygon with rtngls, Pro. 33r Symp. ountions o omputr Sin, 992, pp Igor Pk n J Yng, Tiling simply onnt rgions with rtngls., J. om. Thory, Sr. 20 (203), no. 7, prtmnt o Mthmtis, Northstrn Univrsity, oston, Msshustts 025 -mil rss: myr.ky@husky.nu.u
Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.
Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,
More information0.1. Exercise 1: the distances between four points in a graph
Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in
More informationPlanar Upward Drawings
C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th
More informationCycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!
Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik
More informationGarnir Polynomial and their Properties
Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,
More information, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management
nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o
More informationThe University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008
Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is
More informationGraph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2
Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny
More information1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
More information12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem)
12/3/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 Ciruits Cyl 2 Eulr
More information5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs
Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl
More informationMath 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.
Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right
More informationb. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?
MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth
More informationPresent state Next state Q + M N
Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I
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 informationExam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013
CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or
More informationWhy the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.
Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y
More informationAlgorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph
Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt
More informationConstructive Geometric Constraint Solving
Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC
More information(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely
. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,
More informationOutline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem)
4/25/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 2 Eulr s rig prolm
More informationDesigning A Concrete Arch Bridge
This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr
More informationECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
More informationOutline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)
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 informationSolutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1
Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):
More informationMAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017
MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT
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 informationS i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.
S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths
More informationModule graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura
Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not
More informationTrees as operads. Lecture A formalism of trees
Ltur 2 rs s oprs In this ltur, w introu onvnint tgoris o trs tht will us or th inition o nroil sts. hs tgoris r gnrliztions o th simpliil tgory us to in simpliil sts. First w onsir th s o plnr trs n thn
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 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 information# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.
How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0
More informationOutline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook
More informationGraphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1
CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml
More informationSeven-Segment Display Driver
7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on
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 informationSolutions to Homework 5
Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()
More informationSimilarity Search. The Binary Branch Distance. Nikolaus Augsten.
Similrity Srh Th Binry Brnh Distn Nikolus Augstn nikolus.ugstn@sg..t Dpt. of Computr Sins Univrsity of Slzurg http://rsrh.uni-slzurg.t Vrsion Jnury 11, 2017 Wintrsmstr 2016/2017 Augstn (Univ. Slzurg) Similrity
More informationSteinberg s Conjecture is false
Stinrg s Conjtur is als arxiv:1604.05108v2 [math.co] 19 Apr 2016 Vinnt Cohn-Aa Mihal Hig Danil Král Zhntao Li Estan Salgao Astrat Stinrg onjtur in 1976 that vry planar graph with no yls o lngth our or
More information1. Determine whether or not the following binary relations are equivalence relations. Be sure to justify your answers.
Mth 0 Exm - Prti Prolm Solutions. Dtrmin whthr or not th ollowing inry rltions r quivln rltions. B sur to justiy your nswrs. () {(0,0),(0,),(0,),(,),(,),(,),(,),(,0),(,),(,),(,0),(,),(.)} on th st A =
More informationNefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim
MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07
More informationSection 10.4 Connectivity (up to paths and isomorphism, not including)
Toy w will isuss two stions: Stion 10.3 Rprsnting Grphs n Grph Isomorphism Stion 10.4 Conntivity (up to pths n isomorphism, not inluing) 1 10.3 Rprsnting Grphs n Grph Isomorphism Whn w r working on n lgorithm
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 informationNumbering Boundary Nodes
Numring Bounry Nos Lh MBri Empori Stt Univrsity August 10, 2001 1 Introution Th purpos of this ppr is to xplor how numring ltril rsistor ntworks ffts thir rspons mtrix, Λ. Morovr, wht n lrn from Λ out
More informationCS 461, Lecture 17. Today s Outline. Example Run
Prim s Algorithm CS 461, Ltur 17 Jr Si Univrsity o Nw Mxio In Prim s lgorithm, th st A mintin y th lgorithm orms singl tr. Th tr strts rom n ritrry root vrtx n grows until it spns ll th vrtis in V At h
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 informationCOMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook
More informationCSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review
rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht
More informationProblem solving by search
Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming
More informationGraphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari
Grphs CSC 1300 Disrt Struturs Villnov Univrsity Grphs Grphs r isrt struturs onsis?ng of vr?s n gs tht onnt ths vr?s. Grphs n us to mol: omputr systms/ntworks mthm?l rl?ons logi iruit lyout jos/prosss f
More information12. Traffic engineering
lt2.ppt S-38. Introution to Tltrffi Thory Spring 200 2 Topology Pths A tlommunition ntwork onsists of nos n links Lt N not th st of nos in with n Lt J not th st of nos in with j N = {,,,,} J = {,2,3,,2}
More informationEE1000 Project 4 Digital Volt Meter
Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s
More informationOutline. Binary Tree
Outlin Similrity Srh Th Binry Brnh Distn Nikolus Austn nikolus.ustn@s..t Dpt. o Computr Sins Univrsity o Slzur http://rsrh.uni-slzur.t 1 Binry Brnh Distn Binry Rprsnttion o Tr Binry Brnhs Lowr Boun or
More informationResearch Article On the Genus of the Zero-Divisor Graph of Z n
Intrntionl Journl o Comintoris, Artil ID 7, pgs http://x.oi.org/.1/14/7 Rsrh Artil On th Gnus o th Zro-Divisor Grph o Z n Huong Su 1 n Piling Li 2 1 Shool o Mthmtil Sins, Gungxi Thrs Eution Univrsity,
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationUsing the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas
SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th
More informationWeighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths
Dt Strutur LECTURE Shortt pth lgorithm Proprti of hortt pth Bllmn-For lgorithm Dijktr lgorithm Chptr in th txtook (pp ). Wight grph -- rminr A wight grph i grph in whih g hv wight (ot) w(v i, v j ) >.
More informationAnnouncements. Not graphs. These are Graphs. Applications of Graphs. Graph Definitions. Graphs & Graph Algorithms. A6 released today: Risk
Grphs & Grph Algorithms Ltur CS Spring 6 Announmnts A6 rls toy: Risk Strt signing with your prtnr sp Prlim usy Not grphs hs r Grphs K 5 K, =...not th kin w mn, nywy Applitions o Grphs Communition ntworks
More informationA 4-state solution to the Firing Squad Synchronization Problem based on hybrid rule 60 and 102 cellular automata
A 4-stt solution to th Firing Squ Synhroniztion Prolm s on hyri rul 60 n 102 llulr utomt LI Ning 1, LIANG Shi-li 1*, CUI Shung 1, XU Mi-ling 1, ZHANG Ling 2 (1. Dprtmnt o Physis, Northst Norml Univrsity,
More informationCS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality:
CS2: Grphs Prihr Ch. 4 Rosn Ch. Grphs A olltion of nos n gs Wht n this rprsnt? n A omputr ntwork n Astrtion of mp n Soil ntwork CS2 - Hsh Tls 2 Dirt Grphs Grphs/Ntworks Aroun Us A olltion of nos n irt
More informationC-201 Sheet Bar Measures 1 inch
Janine M. lexander, P.. P.. No. 9, L 0 N. PRK RO, SUIT 0 HOLLYWOO, LORI 0 PHON: (9) - X: (9) 08- No.: 9 I ST SRIPTION Y GT VLVS SHLL RSILINT ST, MNUTUR TO MT OR X TH RQUIRMNTS O WW 09 (LTST RVISION) N
More informationDUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski
Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This
More informationRegister Allocation. Register Allocation. Principle Phases. Principle Phases. Example: Build. Spills 11/14/2012
Rgistr Allotion W now r l to o rgistr llotion on our intrfrn grph. W wnt to l with two typs of onstrints: 1. Two vlus r liv t ovrlpping points (intrfrn grph) 2. A vlu must or must not in prtiulr rhitturl
More informationComplete Solutions for MATH 3012 Quiz 2, October 25, 2011, WTT
Complt Solutions or MATH 012 Quiz 2, Otor 25, 2011, WTT Not. T nswrs ivn r r mor omplt tn is xpt on n tul xm. It is intn tt t mor omprnsiv solutions prsnt r will vlul to stunts in stuyin or t inl xm. In
More informationCSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata
CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl
More informationSAMPLE PAGES. Primary. Primary Maths Basics Series THE SUBTRACTION BOOK. A progression of subtraction skills. written by Jillian Cockings
PAGES Primry Primry Mths Bsis Sris THE SUBTRACTION BOOK A prorssion o sutrtion skills writtn y Jillin Cokins INTRODUCTION This ook is intn to hlp sur th mthmtil onpt o sutrtion in hilrn o ll s. Th mstry
More informationCS September 2018
Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o
More informationGraph Contraction and Connectivity
Chptr 17 Grph Contrtion n Conntivity So r w hv mostly ovr thniqus or solving prolms on grphs tht wr vlop in th ontxt o squntil lgorithms. Som o thm r sy to prllliz whil othrs r not. For xmpl, w sw tht
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More informationThis chapter covers special properties of planar graphs.
Chptr 21 Plnr Grphs This hptr ovrs spil proprtis of plnr grphs. 21.1 Plnr grphs A plnr grph is grph whih n b rwn in th pln without ny gs rossing. Som piturs of plnr grph might hv rossing gs, but it s possibl
More informationFSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *
CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if
More informationNP-Completeness. CS3230 (Algorithm) Traveling Salesperson Problem. What s the Big Deal? Given a Problem. What s the Big Deal? What s the Big Deal?
NP-Compltnss 1. Polynomil tim lgorithm 2. Polynomil tim rution 3.P vs NP 4.NP-ompltnss (som slis y P.T. Um Univrsity o Txs t Dlls r us) Trvling Slsprson Prolm Fin minimum lngth tour tht visits h ity on
More informationChapter 9. Graphs. 9.1 Graphs
Chptr 9 Grphs Grphs r vry gnrl lss of ojt, us to formliz wi vrity of prtil prolms in omputr sin. In this hptr, w ll s th sis of (finit) unirt grphs, inluing grph isomorphism, onntivity, n grph oloring.
More informationGrade 7/8 Math Circles March 4/5, Graph Theory I- Solutions
ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n
More informationProperties of Hexagonal Tile local and XYZ-local Series
1 Proprtis o Hxgonl Til lol n XYZ-lol Sris Jy Arhm 1, Anith P. 2, Drsnmik K. S. 3 1 Dprtmnt o Bsi Sin n Humnitis, Rjgiri Shool o Enginring n, Thnology, Kkkn, Ernkulm, Krl, Ini. jyjos1977@gmil.om 2 Dprtmnt
More informationLast time: introduced our first computational model the DFA.
Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody
More informationMore Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations
Mr Funtins Grphs, Pruts, & Rltins Unirt Grphs An unirt grph is pir f 1. A st f ns 2. A st f gs (whr n g is st f tw ns*) Friy, Sptmr 2, 2011 Ring: Sipsr 0.2 ginning f 0.4; Stughtn 1.1.5 ({,,,,}, {{,}, {,},
More informationModule 2 Motion Instructions
Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr
More informationarxiv: v1 [math.co] 15 Dec 2015
On th Plnr Split Thiknss of Grphs Dvi Eppstin, Philipp Kinrmnn, Stphn Koourov, Giuspp Liott, Ann Luiw, Au Mignn, Djyoti Monl, Hmih Vosoughpour, Su Whitsis 8, n Stphn Wismth 9 rxiv:.89v [mth.co] D Univrsity
More informationXML and Databases. Outline. Recall: Top-Down Evaluation of Simple Paths. Recall: Top-Down Evaluation of Simple Paths. Sebastian Maneth NICTA and UNSW
Smll Pth Quiz ML n Dtss Cn you giv n xprssion tht rturns th lst / irst ourrn o h istint pri lmnt? Ltur 8 Strming Evlution: how muh mmory o you n? Sstin Mnth NICTA n UNSW
More informationPolygons POLYGONS.
Polgons PLYGNS www.mthltis.o.uk ow os it work? Solutions Polgons Pg qustions Polgons Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon f g h Polgon Not polgon Polgon Not polgon Polgon
More informationlearning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms
rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r
More informationPage 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S.
ConTst Clikr ustions Chtr 19 Physis, 4 th Eition Jms S. Wlkr ustion 19.1 Two hrg blls r rlling h othr s thy hng from th iling. Wht n you sy bout thir hrgs? Eltri Chrg I on is ositiv, th othr is ngtiv both
More informationThe Plan. Honey, I Shrunk the Data. Why Compress. Data Compression Concepts. Braille Example. Braille. x y xˆ
h ln ony, hrunk th t ihr nr omputr in n nginring nivrsity of shington t omprssion onpts ossy t omprssion osslss t omprssion rfix os uffmn os th y 24 2 t omprssion onpts originl omprss o x y xˆ nor or omprss
More informationUniform 2D-Monotone Minimum Spanning Graphs
CCCG 2018, Winnipg, Cn, August 8 10, 2018 Uniorm 2D-Monoton Minimum Spnning Grphs Konstntinos Mstks Astrt A gomtri grph G is xy monoton i h pir o vrtis o G is onnt y xy monoton pth. W stuy th prolm o prouing
More informationFormal Concept Analysis
Forml Conpt Anlysis Conpt intnts s losd sts Closur Systms nd Implitions 4 Closur Systms 0.06.005 Nxt-Closur ws dvlopd y B. Gntr (984). Lt M = {,..., n}. A M is ltilly smllr thn B M, if B A if th smllst
More informationUNCORRECTED PAGE PROOFS
7 Shps 9 Unrstning lngth isovr n ojts Msurmnt n gomtry Wht informtion n w gthr from 2 shps n 3 ojts? 7 7 7 7 7E 7F 7G 7H 7I Tringl proprtis Quriltrl proprtis 2 shps n 3 ojts Isomtri rwings n plns Nts n
More informationa b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...
Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th
More informationMultipoint Alternate Marking method for passive and hybrid performance monitoring
Multipoint Altrnt Mrkin mtho or pssiv n hyri prormn monitorin rt-iool-ippm-multipoint-lt-mrk-00 Pru, Jul 2017, IETF 99 Giuspp Fiool (Tlom Itli) Muro Coilio (Tlom Itli) Amo Spio (Politnio i Torino) Riro
More informationOpenMx Matrices and Operators
OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(
More informationarxiv: v1 [cs.ds] 20 Feb 2008
Symposium on Thortil Aspts of Computr Sin 2008 (Borux), pp. 361-372 www.sts-onf.org rxiv:0802.2867v1 [s.ds] 20 F 2008 FIXED PARAMETER POLYNOMIAL TIME ALGORITHMS FOR MAXIMUM AGREEMENT AND COMPATIBLE SUPERTREES
More informationTangram Fractions Overview: Students will analyze standard and nonstandard
ACTIVITY 1 Mtrils: Stunt opis o tnrm mstrs trnsprnis o tnrm mstrs sissors PROCEDURE Skills: Dsriin n nmin polyons Stuyin onrun Comprin rtions Tnrm Frtions Ovrviw: Stunts will nlyz stnr n nonstnr tnrms
More informationThe University of Sydney MATH 2009
T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n
More informationDecimals DECIMALS.
Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths
More informationJonathan Turner Exam 2-10/28/03
CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm
More informationChapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
More information