Welcome to... Problem Analysis and Complexity Theory , 3 VU


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1 Welcome to... Problem Analysis and Complexity Theory , 3 VU Birgit Vogtenhuber Institute for Software Technology office hour: Tuesday 10:30 11:30 slides: Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 1
2 Last Time NPcompleteness exercises: HAMPATH P UHAMPATH HAMCYCLE = P UHAMCYCLE strong NPcompletess of TSP Summary:  picture  proof + reduction checklist Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 2
3 Last Time NPcompleteness exercises: L bold = strongly NPC Arrows HAMPATH indicate P UHAMPATH HAMCYCLE = P UHAMCYCLE SAT ksat for k > 3 presented reductions strong NPcompletess of TSP 3SAT Summary:  picture MAX2SAT SUBSET SUM  proof + reduction checklist HAMPATH INDEP. SET KNAPSACK HAMCYCLE UHAMPATH CLIQUE VERTEX COVER UHAMCYCLE TSP Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 2
4 Last Time New complexity classes via complements: UNSAT: unsatisfiability of boolean formulas Definition of coclasses including conp Discussion P  NP  conp Picture of one possibility Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 2
5 Last Time New complexity classes via complements: UNSAT: unsatisfiability of boolean formulas Definition of coclasses NP including conp conp Discussion P  NP  conp P Picture of one possibility NPC SAT conpc UNSAT Picture: one possibility! Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 2
6 This Time: Space Complexity recall definitions: SPACE(f(n)), NSPACE(f(n)), L, NL, PSPACE clarify computation model and #configurations revisit containment relations of complexity classes two examples: SAT, PATH recall logspace reductions L consider closeness under L define completeness w.r.t. L prove Savitch s Theorem define conl Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 3
7 Definition: Space Complexity: Definition Let s : N N be a complexity function. Deterministic Space: SPACE(s(n)) := {L L is a language decidable by an O(s(n))space det. TM } Nondeterministic Space: NSPACE(s(n)) := {L L is a language decidable by an O(s(n))space nondet. TM } Det. Polynomial Space: PSPACE := k 1 SPACE(nk ) Det. Logarithmic Space: L := SPACE(log n) Nondet. Logarithmic Space: NL := NSPACE(log n) Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 4
8 Space Complexity: Definition Definition: Let s : N N be a complexity function. Deterministic Space: SPACE(s(n)) := {L L is a language decidable PSPACEby an O(s(n))space det. TM } Nondeterministic Space: NSPACE(s(n)) := {L L is a language decidable NL by an Input uses n cells How can O(s(n))space nondet. TM } L a TM use only log n space? Det. Polynomial Space: PSPACE := k 1 SPACE(nk ) Det. Logarithmic Space: L := SPACE(log n) Nondet. Logarithmic Space: NL := NSPACE(log n) Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 4
9 Input / Work / Output TM Model of computation: special 3tape DTM / NTM Input tape: read only! Work tape: only one counted Output tape: write only! no going back Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 5
10 Input / Work / Output TM Model of computation: special 3tape DTM / NTM 1. Q: finite set of states, 2. Σ: finite input alphabet not containing the blank symbol and the start symbol, 3. Γ: finite work alphabet with, Γ and Σ Γ, 4. transition function δ : Q Σ Γ Q {L,, R} in Γ {L,, R} work Γ {, R} out, 5. q 0 Q: start state, 6. q accept Q: accept state, and 7. q reject Q: reject state, where q reject q accept. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 5
11 Input / Work / Output TM Model of computation: special 3tape DTM / NTM 1. Q: finite set of states, 2. Σ: finite input alphabet not containing the blank symbol and the start symbol, 3. Γ: finite work alphabet with, Γ and Σ Γ, 4. transition relation : Q Σ Γ P (Q {L,, R} in Γ {L,, R} work Γ {, R} out ), 5. q 0 Q: start state, 6. q accept Q: accept state, and 7. q reject Q: reject state, where q reject q accept. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 5
12 Input / Work / Output TM Model of computation: special 3tape DTM / NTM Question: How many different configurations for given input of size N and worktape size S? 1. input tape head position: N 2. work tape content: Γ S 3. work tape head position: S 4. the machines state: Q in total: N Γ S S Q input unchangable 1 instead of Σ N! output unreadable need not be counted! Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 5
13 Input / Work / Output TM Model of computation: special 3tape DTM / NTM Question: How many different configurations for given input of size N and worktape size S? 1. input tape head position: N 2. work tape content: Γ S 3. work tape head position: S 4. the machines state: Q in total: Corollary: N Γ S S Q L P, NL NP. Proof: Space bound S = O(log N) N Γ log N log N Q = O(p(N)). Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 5
14 Containment Relations L NL P NP PSPACE EXP L NL, P NP: DTM = special case of NTM PSPACE EXP, L P: # configurations of a DTM P PSPACE: O(f(n))time O(f(n))space EXP PSPACE NP P NL NP PSPACE: simulate O(f(n))time NTM with O(f(n))space DTM NL P: later in this course L Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 6
15 Containment Relations L NL P NP PSPACE EXP L NL, P NP: DTM = special case of NTM PSPACE EXP, L P: # configurations of a DTM P PSPACE: O(f(n))time O(f(n))space EXP PSPACE NP P NL NP PSPACE: simulate O(f(n))time NTM with O(f(n))space DTM NL P: later in this course lesson L Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 6
16 Example: SAT Claim: SAT SPACE(n). M = On input φ, where φ is a Boolean formula: 1. For each truth assignment T of the variables x 1,... x m of φ: 2. Evaluate φ on T 3. If φ(t ) evaluates to true then accept φ 4. reject φ In one iteration of the loop, M needs to store only the truth assignment T O(m) = O(n) space. Each iteration can use the same portion of the work tape M runs in O(n) = linear space in total. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 7
17 Example: PATH In a directed graph G = (V, E), a (simple) path from s V to t V is a directed path that starts at s, ends at t, and visits every vertex of V at most once. Decision problem PATH: Does a given directed graph G contain a path from s to t? t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 8
18 Example: PATH In a directed graph G = (V, E), a (simple) path from s V to t V is a directed path that starts at s, ends at t, and visits every vertex of V at most once. Decision problem PATH: Does a given directed graph G contain a path from s to t? Instance: A directed graph G = (V, E) and s, t V. t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 8
19 Claim: Example: PATH PATH is in NL. nondeterministic algorithm for G = (V, E), s, t: 1. Start with u = s 2. for i = 1,..., V : 3. let u be a nondeterministic neighbor of u 4. if u = t then acccept 5. reject needed Space: O(log V ) for current u O(log V ) for counting to V O(log V ) = O(log n) total Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ t s
20 Input / Work / Witness TM Alternative formulation for spacebounded NTM: Input tape: read only! Work tape: only one counted Witness tape: read only! no going back witness for PATH: a path s t Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/2015 9
21 Definition: LogSpace Reductions A language A is logspace (Karp) reducible to a language B (denoted by A L B), if there exists a logspace computable function f : Σ A Σ B w A f(w) B. (called reductionfunction), such that logspace DTM that outputs f(w) on input w. A f B Σ A \A Σ B \B Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
22 LogSpace Reductions Theorem: L is closed under logspace (Karp) reductions: Claim: If f is a logspace reduction from A to B and B L, then A L. Proof for polynomial time reductions: w a polynomial time algorithm for A polynomialtime algorithm for f f(w) polynomialtime algorithm for B f(w) B? w A? Question: Why does this reasoning not work for L and L? f(w) too large for inside A! Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
23 LogSpace Reductions Theorem: L is closed under logspace (Karp) reductions: Claim: If f is a logspace reduction from A to B and B L, then A L. Proof: Make a machine M A that on input w for A... simulates B s machine M B on input f(w) whenever M B would read the i th symbol of its input f(w), runs f s machine M f on w and waits for the i th symbol to be outputted. Theorem: The complexity classes L, NL, PSPACE, P, NP,and EXP are closed under logspace (Karp) reductions. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
24 LogSpace Reductions Theorem: L is closed under logspace (Karp) reductions: Claim: If f is a logspace reduction from A to B and B L, then A L. Proof: Make a machine M A that on input w for A... simulates B s machine M B on input f(w) whenever M B would read the i th symbol of its input f(w), runs f s machine M f on w and waits for the i th symbol to be outputted. Definition: A decision problem B NL is called NLcomplete if A L B for all A NL. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
25 Theorem: PATH is NLcomplete PATH is NLcomplete. Proof for NLhardness: Given an arbitrary logspace NTM N and an input w, find a logspace computable function f that maps N, w to an instance f(n, w) = (G, s, t) of PATH such that G has path from s to t N accepts w N, w f t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
26 PATH is NLcomplete Configurations graph G N,w = (V, E) for (N, w): vertex set V : all configurations of (N, w) edge (u, v) E iff transition u v possible in (N, w) start vertex s: start configuration of (N, w) target vertex t: accepting configuration of (N, w) Modify N to have a unique accepting configuration s G has path from s to t N accepts w t Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
27 Theorem: PATH is NLcomplete PATH is NLcomplete. Proof for NLhardness: Given an arbitrary logspace NTM N and an input w, find a logspace computable function f that maps N, w to an instance f(n, w) = (G, s, t) of PATH such that G has path from s to t N accepts w N, w f t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
28 PATH is NLcomplete Logspace computability of G N,w for (N, w): V : any configuration can be represented with c log w space (c const.) for all strings s of length s c log w : test whether s is a legal configuration of (N, w) output s iff yes E: try all pairs (c 1, c 2 ) of configurations: test whether with the tape contents and head locations given in c 1, c 2 is a possible next configuration (by the transition relation ) output (c 1, c 2 ) iff yes Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
29 Theorem: PATH is NLcomplete PATH is NLcomplete. Proof for NLhardness: Given an arbitrary logspace NTM N and an input w, find a logspace computable function f that maps N, w to an instance f(n, w) = (G, s, t) of PATH such that G has path from s to t N accepts w N, w f t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
30 PATH is NLcomplete Theorem: Corollary: NL P. PATH is NLcomplete. Proof: PATH is NLcomplete. reduction f from proof in space O(log n). f in time n2 O(log n) = O(p(n)). For any A NL: A P PATH. PATH P For any A NL: A P. t s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
31 Containment Relations L NL P NP PSPACE EXP L NL, P NP: DTM = special case of NTM PSPACE EXP, L P: # configurations of a DTM P PSPACE: O(f(n))time O(f(n))space EXP PSPACE NP P NL NP PSPACE: simulate O(f(n))time NTM with O(f(n))space DTM NL P: later in this lesson PATH NLcomplete and P. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ L
32 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Proof: 1. We construct an algorithm path(g, x, y, i) that returns whether there exists a path in G from x to y that has length at most 2 i. path(g, x, y, i): if i = 0: if x = y or (x, y) E: accept if i > 0: for each z V : if path(g, x, z, i 1) and path(g, z, y, i 1): accept reject Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ t s
33 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Proof: 2. path(g, x, y, log n ) answers the decision question PATH(G, x, y): If there exists a path P from x to y then log n P has length P n 2 P can be split into two halves, both of length log n 1 at most P /2 n/2 2 Two paths P 1 : x z and P 2 : z y can be combined into one path P : x y with length P P 1 + P 2 Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
34 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Proof: 3. path(g, x, y, log n ) can be computed in O((log n) 2 ) space: use stack for recursion with triplets (x, y, i). For each vertex z: write (x, z, i 1) to stack if path(x, z, i 1) accepts replace (x, z, i 1) by (z, y, i 1). else replace (x, z, i 1) by (x, z, i 1). each stack element needs at most space 3 log n as i log n, the height of the stack is at most log n the total stack space is O((log n) 2 ). Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
35 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Corollary: NL = NSPACE(log n) SPACE((log n) 2 ). Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
36 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Corollary (Savitch s Theorem): f Ω(log n) : NSPACE(f(n)) SPACE((f(n)) 2 ). Proof: 1. simulate f(n)space NTM N with input w by running path on its configurations graph G N,w. 2. G N,w is represented only implicitly. It suffices to read w and the description of N for i = G N,w has c f(n) vertices (for some constant c). path needs at most O((log(c f(n) )) 2 ) = O((f(n)) 2 ) space. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
37 Savitch s Theorem Theorem (Savitch 1970): PATH SPACE((log n) 2 ). Corollary (Savitch s Theorem): f Ω(log n) : NSPACE(f(n)) SPACE((f(n)) 2 ). Exercise: Perform path with the shown graph path(g, x, y, i): if i=0: if x = y or (x, y) E: accept t if i > 0: for each z V : if path(g, x, z, i 1) and path(g, z, y, i 1): accept reject Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ s
38 Discussion Space complexity classes: especially L, NL logspace reductions L closeness under L NL P: PATH is NLcomplete NL SPACE((log n) 2 ): PATH SPACE((log n) 2 ) Savitch s Theorem: NSPACE(f(n)) SPACE(f 2 (n)) deterministic simulation of nondeterministic spacebounded computation does not incur that large overhead! Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
39 Next More about space complexity classes: NL vs. conl PSPACE Motivation: Better understanding of space complexity Differences between time complexity and space complexity Start with NOPATH: How complicated is it to decide if a given graph G does not contain a path from s to t? Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
40 NOPATH NOPATH: Nonexistence of a path NOPATH = {(G, s, t) G contains no path from s to t} Decision Problem: Does a given directed graph G not contain a path from s to t? Instance: A directed graph G = (V, E) and s, t V. Question: In which complexity classes is NOPATH for sure? Definition: conl= {A A c NL} s t Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
41 NOPATH NOPATH: Nonexistence of a path NOPATH = {(G, s, t) G contains no path from s to t} Decision Problem: Does a given directed graph G not contain a path from s to t? Instance: A directed graph G = (V, E) and s, t V. Question: In which complexity classes is NOPATH for sure? Answers: P, SPACE(log 2 (n)), conl; NOPATH is conlcomplete s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ t
42 NOPATH NOPATH: Nonexistence of a path NOPATH = {(G, s, t) G contains no path from s to t} Decision Problem: Does a given directed graph G not contain a path from s to t? Instance: A directed graph G = (V, E) and s, t V. Question: What if we can prove that NOPATH is in NL? Answer: Then we have proven NL= conl! s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ t
43 REACHABILITY REACHABILITY: Given a directed graph G and a vertex s of G, how many vertices are reachable from s? Instance: A directed graph G = (V, E) and s V. Output: The number of vertices reachable from s. Example: How many in the shown graph? Question: How complicated is it to compute REACHABILITY? s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
44 REACHABILITY REACHABILITY: Given a directed graph G and a vertex s of G, how many vertices are reachable from s? Instance: A directed graph G = (V, E) and s V. Output: The number of vertices reachable from s. Example: How many in the shown graph? Question: How complicated is it to compute REACHABILITY? s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
45 REACHABILITY Theorem (ImmermanSzelepscényi 1988): Given a directed graph G with n vertices and a vertex s of G, the number of vertices that are reachable from s can be computed in NSPACE(log n) = NL. Proof: Let S(k) be the set of vertices of G that are reachable from s on a path of length at most k. S(n 1) is the total number of vertices that are reachable from s. We iteratively compute S(1), S(2),..., S(n 1) with a nondeterministic algorithm using space O(log n) Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
46 REACHABILITY countreachablenodes(g, s) c prev = 1 for k = 1 to n 1 /* c prev : # of vertices reachable in k 1 steps */ c curr = 0 for each v V /* Is v reachable from s in k steps? */ if iselement(s, v, G, k, c prev ): c curr = c curr + 1 c prev = c curr output c prev Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
47 REACHABILITY iselement(s, v, G, k, c prev ) sc = 0, reply = FALSE find path s k 1 w 1 v for each w V goodp ath = TRUE, u curr = s for j = 1 to k 1 /* Guess a k 1 long path */ u prev = u curr, guess u curr if (u prev, u curr ) E: goodp ath = FALSE if u curr == w: break /* check if path is good and ends in w */ if goodp ath and u curr == w sc = sc + 1 if (w, v) E: reply = TRUE if sc < c prev : nondeterministic abort else return reply Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
48 REACHABILITY iselement(s, v, G, k, c prev ): guesses a path of length k 1 for each vertex w. counts vertices w for which it guessed a path checks that it found all (sc == c prev ): some computation branch finds all (first) checks for all w with path whether it can be extended with an edge to v extendible for one w implies answer TRUE countreachablenodes(g, s) iterative computation of c prev enables sanitycheck for iselement(s, v, G, k, c prev ) stores only constantly many variables of size log n runs in NSPACE(log n) Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
49 REACHABILITY Theorem (ImmermanSzelepscényi 1988): Given a directed graph G with n vertices and a vertex s of G, the number of vertices that are reachable from s can be computed in NSPACE(log n) = NL. Proof: Let S(k) be the set of vertices of G that are reachable from s on a path of length at most k. S(n 1) is the total number of vertices that are reachable from s. We iteratively compute S(1), S(2),..., S(n 1) with a nondeterministic algorithm using space O(log n) Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
50 NOPATH is in NL NOPATH: Nonexistence of a path NOPATH = {(G, s, t) G contains no path from s to t} Decision Problem: Does a given directed graph G not contain a path from s to t? Instance: A directed graph G = (V, E) and s, t V. Question: What if we can prove that NOPATH is in NL? Answer: Then we have proven NL= conl! s Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/ t
51 NOPATH is in NL Corollary: from ImmermanSzelepscényi it follows that NOPATH is a member of NL Proof by algorithm: unreachability(g, s, t) c prev = 1 for k = 1 to n 1 /* c prev : # of vertices reachable in k 1 steps */ c curr = 0 for each v V /* Is v reachable from s in k steps? */ if iselement(s, v, G, k, c prev ): c curr = c curr + 1 if v == t: REJECT c prev = c curr ACCEPT Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
52 NOPATH is in NL Corollary: Corollary: Corollary: Proof: from ImmermanSzelepscényi it follows that NOPATH is a member of NL NL= conl the ImmermanSzelepscényi theorem: Let f(n) be an arbitrary complexity function with f(n) = Ω(log n). Then it holds that NSPACE(f(n)) = conspace(f(n)). Exercise. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
53 Remarks on L, NL, and conl PATH and NOPATH are both NLcomplete. 2SAT (and 2UNSAT) are NLcomplete as well. Exercise: Prove that 2UNSAT NL. It is generally believed that NP conp and the same also appeared to hold for NL and conl. The ImmermanSzelepscényi theorem is one of the most surprising ones concerning the relationships of complexity classes, showing that the intuition about computation has still many gaps! It still remains open whether L NL. Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
54 Remarks on L, NL, and conl PATH and NOPATH are both NLcomplete. 2SAT (and 2UNSAT) are NLcomplete as well. Exercise: Prove that 2UNSAT NL. It is generally believed that NP conp and the same also appeared to hold for NL and conl. The ImmermanSzelepscényi theorem is one of the most surprising ones concerning the relationships of complexity classes, showing that the intuition about computation has still many gaps! It still remains open whether L NL. Thank you for your attention! Birgit Vogtenhuber Problem Analysis and Complexity Theory, , winter term 2014/
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