Math Revew CptS 223 dvanced Data Structures Larry Holder School of Electrcal Engneerng and Computer Scence Washngton State Unversty 1
Why do we need math n a data structures course? nalyzng data structures and algorthms Dervng formulae for tme and memory requrements Wll the soluton scale? Provng algorthm correctness 2
Floors and Celngs floor(x), denoted x, s the greatest nteger x celng(x), denoted x, s the smallest nteger x ormally used to dvde nput nto ntegral parts 2 + 2 3
Example Consder lgorthm1 that dvdes the nput array n half and calls lgorthm2 on each half lgorthm1 (,n) // s an nteger array of sze n x floor(n/2) lgorthm2 (,1,x) lgorthm2 (,x+1,n) ssume lgorthm2(,,j) s runnng tme s proportonal to (j-+1) What s the runnng tme of lgorthm1? 4
Exponents 5 1 2 2 2 2 2 ) ( + + + + B B B B B B
Logarthms log log log log log log lg ln B B log log B < B log log log log B log 2 for all ; ; e, B, C B log + log B; B e C C B B > 0 2.7182... "logarthm of > 0, 1, B > 0 B base " "natural logarthm" In Wess book, log log 2 6
Example How many tmes to halve an array of length n untl ts length s 1? Halve (n) 0 whle n 1 + 1 n floor(n/2) return 7
Factorals n! n! < n n! 1 f n 0 n ( n 1)! f n > 0 n n 2π n ( n / e) (1 +θ (1/ n)) Strlng's approxmaton PermutatonSort (,n) // s an nteger array of length n whle s not n order Permute () 8
9 Seres General Lnearty rthmetc seres + + + f f f f 0 ) (... (1) (0) ) ( + 1 2 1) ( + + g f c g cf 0 0 0 ) ( ) ( )) ( ) ( (
Seres Geometrc seres 0 0 + 1 1 1 0 1 ; 1 f 0 < < 1 Example How many nodes n a complete bnary tree of depth D? 3 15 10 22 12 19 27 10
Modular rthmetc mod ( mod ) " s congruent to B modulo " E.g., 81 If Then + C and D 61 1(mod10) B (mod ) / BD (mod ) ( B mod ) B + C (mod ) "remander" B (mod ) Bass of most encrypton schemes: (Message mod Key) 11
Proofs What do we want to prove? Propertes of a data structure always hold for all operatons lgorthm runnng tme/memory wll never exceed some lmt lgorthm wll always be correct lgorthm wll always termnate Technques Proof by nducton Proof by counterexample Proof by contradcton 12
Proof by Inducton Goal: Prove some hypothess s true Three-step process 1. Base case: Show hypothess s true for some ntal condtons 2. Inductve hypothess: ssume hypothess s true for all values k 3. Show hypothess s true for next larger value (typcally k+1) 13
Example Prove arthmetc seres ( + 1) 1 2 (Step 1) Base case: Show true for 1 1 1(1 + 1) 1 1 2 14
Example (cont.) (Step 2) ssume true for k (Step 3) Show true for k+1 k + 1 1 ( k + 1) + 1 k( k + 1) ( k + 1) + 2 2( k + 1) + k( k + 1) 2 ( k + 1)( k + 2) 2 k 15
More Examples Prove the geometrc seres 0 + 1 1 1 Prove that the number of nodes n a complete bnary tree of depth D s 2 D+1-1 16
Proof by Counterexample Prove hypothess s not true by gvng an example that doesn t work Example: 2 > 2? Proof by example? Proof by lots of examples? Proof by all possble examples? Emprcal proof Hard when nput sze and contents can vary arbtrarly 17
nother Example 100 D 20 100 10 10 Travelng salesman problem Gven ctes and costs for travelng between each par of ctes, fnd the least-cost tour to vst each cty exactly once Hypothess B C 10 Gven a least-cost tour for ctes, the same tour wll be least-cost for (-1) ctes E.g., f B C D s the least-cost tour for ctes {,B,C,D}, then B C wll be the least-cost tour for ctes {,B,C} 18
nother Example (cont.) Counterexample Cost ( B C D) 40 (optmal) Cost ( B C) 30 Cost ( C B) 20 100 D 20 100 10 10 B C 10 19
Proof by Contradcton ssume hypothess s false Show ths assumpton leads to a contradcton (.e., some know property s volated) Can t use specal cases or specfc examples Therefore, hypothess must be true 20
Example Varant of travelng salesman problem Gven ctes and costs for travelng between each par of ctes, fnd the least-cost path to go from cty to cty Y Hypothess 100 D 20 100 100 10 least-cost path from to Y contans least-cost paths from to every cty on the path E.g., f C1 C2 C3 Y s the least-cost path from to Y, then B C 10 C1 C2 C3 s the least-cost path from to C3 C1 C2 s the least-cost path from to C2 C1 s the least-cost path from to C1 21
Example (cont.) ssume hypothess s false I.e., Gven a least-cost path P from to Y that goes through C, there s a better path P from to C than the one n P Show a contradcton But we could replace the subpath from to C n P wth ths lesser-cost path P The path cost from C to Y s the same Thus we now have a better path from to Y But ths volates the assumpton that P s the least-cost path from to Y Therefore, the orgnal hypothess must be true P C Y 22
Recurson recursve functon s defned n terms of tself 1 f n 0 Example: n! n ( n 1)! f n > 0 Factoral (n) f n 0 then return 1 else return (n * Factoral (n-1)) 23
Basc Rules of Recurson Base cases Must always have some base cases, whch can be solved wthout recurson Makng progress Recursve calls must always make progress toward a base case Desgn rule ssume all recursve calls work Compound nterest rule ever duplcate work by solvng the same nstance of a problem n separate recursve calls
Example Fbonacc numbers F(0) 1 F(1) 1 F(n) F(n-1) + F(n-2) Fbonacc (n) f (n 1) then return 1 else return (Fbonacc (n-1) + Fbonacc (n-2)) Dd you know? The number of petals on a flower s usually a Fbonacc number? Ths dasy has F(8)34 petals. 25
Example (cont.) Fbonacc (5) F(5) F(4) F(3) F(3) F(2) F(2) F(1) F(2) F(1) F(1) F(0) F(1) F(0) F(1) F(0) 26
Example (cont.) Show that the runnng tme T(n) of Fbonacc(n) s exponental n n Use mathematcal nducton Show T(n) c2 n for some postve constant c Base case: T(1) c2 n 2c (true for some constant c > 0) Inductve hypothess: T(k) c2 k Show T(k+1) c2 k+1 ctually, only proved that T(n) s no more than exponental eed to also prove T(n) s at least exponental 27
Summary Floors, celngs, exponents, logarthms, seres, and modular arthmetc Proofs by mathematcal nducton, counterexample and contradcton Recurson Tools to help us analyze the performance of our data structures and algorthms 28