13. RANDOMIZED ALGORITHMS

Transcription

1 Randomization Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley Algorithmic design patterns. Greedy. Divide-and-conquer. Dynamic programming. Network flow. Randomization. Randomization. Allow fair coin flip in unit time. Why randomize? Can lead to simplest, fastest, or only known algorithm for a particular problem. in practice, access to a pseudo-random number generator Ex. Symmetry-breaking protocols, graph algorithms, quicksort, hashing, load balancing, Monte Carlo integration, cryptography. Last updated on 3/2/8 4:57 PM 2 Contention resolution in a distributed system Contention resolution. Given n processes P,, P n, each competing for access to a shared database. If two or more processes access the database simultaneously, all processes are locked out. Devise protocol to ensure all processes get through on a regular basis. Restriction. Processes can t communicate. Challenge. Need symmetry-breaking paradigm. P P 2... P n 4

2 Contention resolution: randomized protocol Contention resolution: randomized protocol Protocol. Each process requests access to the database at time t with probability p = /n. Claim. The probability that process i fails to access the database in en rounds is at most / e. After e n (c ln n) rounds, the probability n -c. Claim. Let S[i, t] = event that process i succeeds in accessing the database at time t. Then / (e n) Pr [S(i, t)] /(2n). Pf. By independence, Pr [S(i, t)] = p ( p) n. process i requests access none of remaining n- processes request access Pf. Let F[i, t] = event that process i fails to access database in rounds through t. By independence and previous claim, we have Pr [F[i, t]] ( /(en)) t. Choose t = e n : Pr[F(i, t)] ( en ) en $ % ( en) en e Setting p = /n, we have Pr [S(i, t)] = /n ( /n) n. value that maximizes Pr[S(i, t)] between /e and /2 Choose t = e n c ln n : Pr[F(i, t)] ( e ) c ln n = n c Useful facts from calculus. As n increases from 2, the function: ( /n) n - converges monotonically from /4 up to / e. ( /n) n converges monotonically from /2 down to / e. 5 6 Contention resolution: randomized protocol Claim. The probability that all processes succeed within 2e n ln n rounds is / n. Pf. Let F[t] = event that at least one of the n processes fails to access database in any of the rounds through t. [ ] = Pr F[i, t ] Pr F[t] " # $ n i= % &' Pr[ F[i, t]] n Choosing t = 2 en c ln n yields Pr[F[t]] n n -2 = / n. n i= union bound previous slide ( ) t en Union bound. Given events E,, E n, " n Pr # $ E i i= % &' n Pr[E i ] i= 7

3 Global minimum cut Contraction algorithm Global min cut. Given a connected, undirected graph G = (V, E), find a cut (A, B) of minimum cardinality. Applications. Partitioning items in a database, identify clusters of related documents, network reliability, network design, circuit design, TSP solvers. Network flow solution. Replace every edge (u, v) with two antiparallel edges (u, v) and (v, u). Pick some vertex s and compute min s- v cut separating s from each other vertex v V. False intuition. Global min-cut is harder than min s-t cut. Contraction algorithm. [Karger 995] Pick an edge e = (u, v) uniformly at random. Contract edge e. - replace u and v by single new super-node w - preserve edges, updating endpoints of u and v to w - keep parallel edges, but delete self-loops Repeat until graph has just two nodes u and v. Return the cut (all nodes that were contracted to form v ). a b c u d v contract u-v w f e a f b c 9 0 Contraction algorithm Contraction algorithm. [Karger 995] Pick an edge e = (u, v) uniformly at random. Contract edge e. - replace u and v by single new super-node w - preserve edges, updating endpoints of u and v to w - keep parallel edges, but delete self-loops Repeat until graph has just two nodes u and v. Return the cut (all nodes that were contracted to form v ). Contraction algorithm Claim. The contraction algorithm returns a min cut with prob 2 / n 2. Pf. Consider a global min-cut (A*, B*) of G. Let F* be edges with one endpoint in A* and the other in B*. Let k = F* = size of min cut. In first step, algorithm contracts an edge in F* probability k / E. Every node has degree k since otherwise (A*, B*) would not be a min-cut E ½ k n k / E 2 / n. Thus, algorithm contracts an edge in F* with probability 2 / n. A* B* F* Reference: Thore Husfeldt 2

4 Contraction algorithm Contraction algorithm Claim. The contraction algorithm returns a min cut with prob 2 / n 2. Pf. Consider a global min-cut (A*, B*) of G. Let F* be edges with one endpoint in A* and the other in B*. Let k = F* = size of min cut. Let Gʹ be graph after j iterations. There are nʹ = n j supernodes. Suppose no edge in F* has been contracted. The min-cut in Gʹ is still k. Since value of min-cut is k, Eʹ ½ k nʹ k / Eʹ 2 / nʹ. Thus, algorithm contracts an edge in F* with probability 2 / nʹ. Let E j = event that an edge in F* is not contracted in iteration j. Pr[E E 2! E n 2 ] = Pr[E ] Pr[E 2 E ]! Pr[E n 2 E E 2! E n 3 ] = ( 2 n) ( n 2 )! ( 2 4) ( 2 3) n 2 n 3 n n ( ) ( 3 ) ( ) ( )! 2 4 = 2 n(n ) 2 n 2 3 Amplification. To amplify the probability of success, run the contraction algorithm many times. Claim. If we repeat the contraction algorithm n 2 ln n times, then the probability of failing to find the global min-cut is / n 2. Pf. By independence, the probability of failure is at most # 2 & % $ n 2 ( ' n 2 lnn ) # = 2 & + % $ n 2 ( * + ' 2 n2,. -. 2ln n ( e ) 2lnn = n 2 ( /x) x /e with independent random choices, 4 Contraction algorithm: example execution Global min cut: context trial Remark. Overall running time is slow since we perform Θ(n 2 log n) iterations and each takes Ω(m) time. trial 2 trial 3 trial 4 Improvement. [Karger Stein 996] O(n 2 log 3 n). Early iterations are less risky than later ones: probability of contracting an edge in min cut hits 50% when n / 2 nodes remain. Run contraction algorithm until n / 2 nodes remain. Run contraction algorithm twice on resulting graph and return best of two cuts. trial 5 (finds min cut) Extensions. Naturally generalizes to handle positive weights. Best known. [Karger 2000] O(m log 3 n). trial 6 faster than best known max flow algorithm or deterministic global min cut algorithm... Reference: Thore Husfeldt 5 6

5 Expectation Expectation. Given a discrete random variable X, its expectation E[X] is defined by: E[X] = j Pr[X = j] Waiting for a first success. Coin is heads with probability p and tails with probability p. How many independent flips X until first heads? E[X] = j Pr[X = j] = j ( p) j p p = p j tails head j ( p) j = p p p = p 2 p 8 Expectation: two properties Guessing cards Useful property. If X is a 0/ random variable, E[X] = Pr[X = ]. Pf. E[X] = j Pr[X = j] = j Pr[X = j] = Pr[X =] not necessarily independent Linearity of expectation. Given two random variables X and Y defined over the same probability space, E[X + Y] = E[X] + E[Y]. Benefit. Decouples a complex calculation into simpler pieces. Game. Shuffle a deck of n cards; turn them over one at a time; try to guess each card. Memoryless guessing. No psychic abilities; can t even remember what s been turned over already. Guess a card from full deck uniformly at random. Claim. The expected number of correct guesses is. Pf. [ surprisingly effortless using linearity of expectation ] Let X i = if i th prediction is correct and 0 otherwise. Let X = number of correct guesses = X + + X n. E[X i ] = Pr[X i = ] = / n. E[X] = E[X ] + + E[X n ] = / n + + / n =. linearity of expectation 9 20

6 Guessing cards Coupon collector Game. Shuffle a deck of n cards; turn them over one at a time; try to guess each card. Guessing with memory. Guess a card uniformly at random from cards not yet seen. Claim. The expected number of correct guesses is Θ(log n). Pf. Let X i = if i th prediction is correct and 0 otherwise. Let X = number of correct guesses = X + + X n. E[X i ] = Pr[X i = ] = / (n ( i )). E[X] = E[X ] + + E[X n ] = / n + + / 2 + / = H(n). Coupon collector. Each box of cereal contains a coupon. There are n different types of coupons. Assuming all boxes are equally likely to contain each coupon, how many boxes before you have coupon of each type? Claim. The expected number of steps is Θ(n log n). Pf. Phase j = time between j and j + distinct coupons. Let X j = number of steps you spend in phase j. Let X = number of steps in total = X 0 + X + + X n. n E[X] = E[X j ] = n n n = n n j i= i = n H(n) linearity of expectation ln(n+) < H(n) < + ln n prob of success = (n j) / n expected waiting time = n / (n j) 2 22 Maximum 3-satisfiability exactly 3 distinct literals per clause Maximum 3-satisfiability. Given a 3-SAT formula, find a truth assignment that satisfies as many clauses as possible. C = x 2 x 3 x 4 C 2 = x 2 x 3 x 4 C 3 = x x 2 x 4 C 4 = x x 2 x 3 C 5 = x x 2 x 4 Remark. NP-hard search problem. Simple idea. Flip a coin, and set each variable true with probability ½, independently for each variable. 24

7 Maximum 3-satisfiability: analysis The probabilistic method Claim. Given a 3-SAT formula with k clauses, the expected number of Corollary. For any instance of 3-SAT, there exists a truth assignment that clauses satisfied by a random assignment is 7k / 8. satisfies at least a 7/8 fraction of all clauses. " if clause C j is satisfied Zj = # $ 0 otherwise. Pf. Consider random variable Pf. Random variable is at least its expectation some of the time. satisfied by random assignment. Let Z = number of clauses k E[Z ] = linearity of expectation E[Z j ] Probabilistic method. [Paul Erdös] Prove the existence of a non-obvious j= k property by showing that a random construction produces it with = Pr[clause C j is satisfied] = j= 7k 8 positive probability! Maximum 3-satisfiability: analysis Maximum 3-satisfiability: analysis Q. Can we turn this idea into a 7/8-approximation algorithm? Johnson s algorithm. Repeatedly generate random truth assignments until A. Yes (but a random variable can almost always be below its mean). one of them satisfies 7k / 8 clauses. Lemma. The probability that a random assignment satisfies 7k / 8 clauses Theorem. Johnson s algorithm is a 7/8-approximation algorithm. is at least / (8k). Pf. By previous lemma, each iteration succeeds with probability / (8k). Pf. Let pj be probability that exactly j clauses are satisfied; By the waiting-time bound, the expected number of trials to find the let p be probability that 7k / 8 clauses are satisfied. satisfying assignment is at most 8k. 7k 8 = E[Z ] = j pj j 0 = j pj + j < 7k /8 j pj j 7k /8 ( 7k 8 ) p j + k p j 8 j < 7k /8 j 7k /8 ( 87 k 8 ) + k p Rearranging terms yields p / (8k)

8 Maximum satisfiability Extensions. Allow one, two, or more literals per clause. Find max weighted set of satisfied clauses. Theorem. [Asano Williamson 2000] There exists a approximation algorithm for MAX-SAT. Monte Carlo vs. Las Vegas algorithms Monte Carlo. Guaranteed to run in poly-time, likely to find correct answer. Ex: Contraction algorithm for global min cut. Las Vegas. Guaranteed to find correct answer, likely to run in poly-time. Ex: Randomized quicksort, Johnson s MAX-3-SAT algorithm. Theorem. [Karloff Zwick 997, Zwick+computer 2002] There exists a 7/8- approximation algorithm for version of MAX-3-SAT in which each clause has at most 3 literals. Theorem. [Håstad 997] Unless P = NP, no ρ-approximation algorithm for MAX-3-SAT (and hence MAX-SAT) for any ρ > 7/8. Remark. Can always convert a Las Vegas algorithm into Monte Carlo, but no known method (in general) to convert the other way. stop algorithm after a certain point very unlikely to improve over simple randomized algorithm for MAX-3-SAT RP and ZPP RP. [Monte Carlo] Decision problems solvable with one-sided error in poly-time. One-sided error. If the correct answer is no, always return no. If the correct answer is yes, return yes with probability ½. ZPP. [Las Vegas] Decision problems solvable in expected poly-time. Theorem. P ZPP RP NP. Fundamental open questions. To what extent does randomization help? Does P = ZPP? Does ZPP = RP? Does RP = NP? can decrease probability of false negative to 2-00 by 00 independent repetitions running time can be unbounded, but fast on average 3

9 Dictionary data type Hashing Dictionary. Given a universe U of possible elements, maintain a subset S U so that inserting, deleting, and searching in S is efficient. Dictionary interface. create(): initialize a dictionary with S =. insert(u): add element u U to S. delete(u): delete u from S (if u is currently in S). lookup(u): is u in S? Hash function. h : U { 0,,, n }. Hashing. Create an array a of length n. When processing element u, access array element a[h(u)]. birthday paradox Collision. When h(u) = h(v) but u v. A collision is expected after Θ( n) random insertions. Separate chaining: a[i] stores linked list of elements u with h(u) = i. Challenge. Universe U can be extremely large so defining an array of size U is infeasible. a[0] a[] jocularly null seriously Applications. File systems, databases, Google, compilers, checksums P2P networks, associative arrays, cryptography, web caching, etc. a[2] suburban untravelled considerating a[n-] browsing Ad-hoc hash function Ad-hoc hash function. int hash(string s, int n) { int hash = 0; for (int i = 0; i < s.length(); i++) hash = (3 * hash) + s[i]; return hash % n; } hash function à la Java string library Deterministic hashing. If U n 2, then for any fixed hash function h, there is a subset S U of n elements that all hash to same slot. Thus, Θ(n) time per lookup in worst-case. Algorithmic complexity attacks When can t we live with ad-hoc hash function? Obvious situations: aircraft control, nuclear reactor, pace maker,. Surprising situations: denial-of-service attacks. malicious adversary learns your ad-hoc hash function (e.g., by reading Java API) and causes a big pile-up in a single slot that grinds performance to a halt Real world exploits. [Crosby Wallach 2003] Linux kernel: save files with carefully chosen names. Perl 5.8.0: insert carefully chosen strings into associative array. Bro server: send carefully chosen packets to DOS the server, using less bandwidth than a dial-up modem. Q. But isn t ad-hoc hash function good enough in practice? 35 36

10 Hashing performance Ideal hash function. Maps m elements uniformly at random to n hash slots. Running time depends on length of chains. Average length of chain = α = m / n. Choose n m expect O() per insert, lookup, or delete. Universal hashing (Carter Wegman 980s) A universal family of hash functions is a set of hash functions H mapping a universe U to the set { 0,,, n } such that For any pair of elements u v : Pr h H h(u) = h(v) Can select random h efficiently. Can compute h(u) efficiently. [ ] /n chosen uniformly at random Challenge. Explicit hash function h that achieves O() per operation. Approach. Use randomization for the choice of h. adversary knows the randomized algorithm you re using, but doesn t know random choice that the algorithm makes Ex. U = { a, b, c, d, e, f }, n = 2. a b c d e f h (x) h 2 (x) H = {h, h 2 } Pr h H [h(a) = h(b)] = /2 Pr h H [h(a) = h(c)] = Pr h H [h(a) = h(d)] = 0... not universal h (x) h 2 (x) h 3 (x) h 4 (x) a b c d e f H = {h, h 2, h 3, h 4 } Pr h H [h(a) = h(b)] = /2 Pr h H [h(a) = h(c)] = /2 Pr h H [h(a) = h(d)] = /2 Pr h H [h(a) = h(e)] = /2 Pr h H [h(a) = h(f)] = 0 universal Universal hashing: analysis Designing a universal family of hash functions Proposition. Let H be a universal family of hash functions mapping a universe U to the set { 0,,, n }; let h H be chosen uniformly at random from H; let S U be a subset of size at most n; and let u S. Then, the expected number of items in S that collide with u is at most. Modulus. We will use a prime number p for the size of the hash table. Integer encoding. Identify each element u U with a base-p integer of r digits: x = (x, x 2,, x r ). Pf. For any s S, define random variable X s = if h(s) = h(u), and 0 otherwise. Let X be a random variable counting the total number of collisions with u. Hash function. Let A = set of all r-digit, base-p integers. For each a = (a, a 2,, a r ) where 0 a i < p, define E h H [X] = E[ s S X s ] = s S E[X s ] = s S Pr[X s =] s S n = S n h a (x) = # r & % a i x i ( mod p maps universe U to set { 0,,, p } $ ' i= linearity of expectation Xs is a 0 random variable universal Hash function family. H = { h a : a A }. Q. OK, but how do we design a universal class of hash functions? 39 40

11 Designing a universal family of hash functions Number theory fact Theorem. H = { h a : a A } is a universal family of hash functions. Pf. Let x = (x, x 2,, x r ) and y = (y, y 2,, y r ) be two distinct elements of U. We need to show that Pr[h a (x) = h a (y)] / p. Since x y, there exists an integer j such that x j y j. We have h a (x) = h a (y) iff a j ( y j x j )!# "# $ z = a i (x i y i ) mod p i j!# "## $ Can assume a was chosen uniformly at random by first selecting all coordinates a i where i j, then selecting a j at random. Thus, we can assume a i is fixed for all coordinates i j. Since p is prime, a j z m mod p has at most one solution among p possibilities. see lemma on next slide Thus Pr[h a (x) = h a (y)] / p. m Fact. Let p be prime, and let z 0 mod p. Then α z m mod p has at most one solution 0 α < p. Pf. Suppose 0 α < p and 0 α 2 < p are two different solutions. Then (α α 2) z 0 mod p; hence (α α 2) z is divisible by p. Since z 0 mod p, we know that z is not divisible by p. It follows that (α α 2) is divisible by p. This implies α = α 2. Bonus fact. Can replace at most one with exactly one in above fact. Pf idea. Euclid s algorithm. here s where we use that p is prime 4 42 Universal hashing: summary Goal. Given a universe U, maintain a subset S U so that insert, delete, and lookup are efficient. Universal hash function family. H = { h a : a A }. h a (x) = # r & % a i x i ( mod p $ ' i= Choose p prime so that m p 2m, where m = S. Fact: there exists a prime between m and 2m. Consequence. Space used = Θ(m). Expected number of collisions per operation is O() time per insert, delete, or lookup. can find such a prime using another randomized algorithm (!) 43

12 Chernoff Bounds (above mean) Chernoff Bounds (above mean) Theorem. Suppose X,, X n are independent 0- random variables. Let X = X + + X n. Then for any µ E[X] and for any δ > 0, we have Pf. [ continued ] Let p i = Pr [X i = ]. Then, sum of independent 0- random variables is tightly centered on the mean Combining everything: for any α 0, +α e α Pf. We apply a number of simple transformations. For any t > 0, Pr[X > (+ δ)µ] = Pr e t X > e t(+δ)µ [ ] e t(+δ)µ E[e tx ] Pr[X > (+ δ)µ] e t(+δ)µ E[e t X i i ] e t(+δ)µ e p i (e t ) i e t(+δ)µ e µ(et ) previous slide inequality above i p i = E[X] µ f(x) = e tx is monotone in x Markov s inequality: Pr[X > a] E[X] / a Now E[e tx ] = E[e t X i i ] = E[e t X i ] i Finally, choose t = ln( + δ). definition of X independence Chernoff Bounds (below mean) Theorem. Suppose X,, X n are independent 0- random variables. Let X = X + + X n. Then for any µ E [X ] and for any 0 < δ <, we have Pf idea. Similar. Remark. Not quite symmetric since only makes sense to consider δ <. 47

13 Load balancing Load balancing Load balancing. System in which m jobs arrive in a stream and need to be processed immediately on m identical processors. Find an assignment that balances the workload across processors. Centralized controller. Assign jobs in round-robin manner. Each processor receives at most m / n jobs. Decentralized controller. Assign jobs to processors uniformly at random. How likely is it that some processor is assigned too many jobs? Analysis. Let X i = number of jobs assigned to processor i. Let Y ij = if job j assigned to processor i, and 0 otherwise. We have E[Y ij ] = /n. Thus, X i = j Y i j, and μ = E[X i ] =. Applying Chernoff bounds with δ = c yields Let γ(n) be number x such that x x = n, and choose c = e γ(n). Union bound with probability /n no processor receives more than e γ(n) = Θ(log n / log log n) jobs. Bonus fact: with high probability, some processor receives Θ(logn / log log n) jobs Load balancing: many jobs Theorem. Suppose the number of jobs m = 6 n ln n. Then on average, each of the n processors handles μ = 6 ln n jobs. With high probability, every processor will have between half and twice the average load. Pf. Let X i, Y ij be as before. Applying Chernoff bounds with δ = yields Pr[X i > 2µ] < e 4 6n ln n < e ln n = n 2 Pr X i < 2 µ < e 2 ( 2 )2 6n ln n = n 2 Union bound every processor has load between half and twice the average with probability 2/n. 5