PSPACE, NPSPACE, L, NL, Savitch's Theorem. More new problems that are representa=ve of space bounded complexity classes

Transcription

1 PSPACE, NPSPACE, L, NL, Savitch's Theorem More new problems that are representa=ve of space bounded complexity classes

2 Outline for today How we'll count space usage Space bounded complexity classes New problems that fit well in these classes Rela=onships between space and =me bounded complexity classes Savitch's Theorem and the power of thinking nondeterminis=cally

3 Coun=ng Space Usage Dis=nguish between a read-only input tape and work tapes of a Turing Machine (TM) Let L be decided by a determinis=c TM (assume that M always halts) L is in SPACE(s(n)) iff there is a constant c such M s work tape heads visit at most c.s(n) cells on inputs of length n NSPACE(s(n)): replace determinis=c by nondeterminis=c

4 Space Bounded Complexity Classes PSPACE = c>0 SPACE(n c ) L = SPACE(log n)

5 True Quan=fied Boolean Formulas (TQBF) w x y z (w x y) ( w x) (x y z) z Given a quan4fied Boolean formula φ, is it true? TQBF is in PSPACE why? Can you bound its =me complexity?

6 Addi=on Given binary numbers x, y and z, is x + y = z? Addi=on is in L (i.e., log space) why? Can you bound its =me complexity?

7 Configura=on Graph of TM M on input x Nodes represent configura=ons of M on x, edges represent transi=ons Assume wlog that the graph has no cycles If M is s(n)-space bounded, can you bound the number of nodes of the graph when x = n?

8 Rela=ng Time and Space PSPACE EXP: a poly(n)-space bounded TM has at most an exponen=al number of different configura=ons, and so must halt aaer an exponen=al number of steps Similarly, L P

9 Rela=ng Time and Space Decidable EXP PSPACE TQBF NP P L Addi=on

10 Directed Graph Reachability (DGR) Given a directed graph G = (V,E) and two nodes s and f, is f reachable from s? DGR is in P use breadth-first or depth-first search but these algorithms use polynomial space too. Is there a more space-efficient solu=on? What if you can use nondeterminism?

11 Directed Graph Reachability (DGR) Given a directed graph G = (V,E) and two nodes s and f, is f reachable from s? DGR is in nondeterminis4c log space: simply guess a path from s to f and verify that each edge of the path is in the graph To save space, store only the current node along the path

12 More Space Bounded Complexity Classes PSPACE = c>0 SPACE(n c ) NPSPACE = c>0nspace(n c ) L = SPACE(log n) NL = NSPACE(log n)

13 More on Rela=ng Time and Space NPSPACE EXP: given NTM M and input x, write down the configura=on graph of M on x; size of the graph is at most exponen=al check if an accep=ng configura=on can be reached from the ini=al configura=on Similarly, NL P

14 Savitch s Theorem: NPSPACE = PSPACE Proof idea: Let L be accepted by NTM M in at most c.s(n) space. We'll describe a determinis=c algorithm A that accepts L in O(s(n) 2 ) space. On input x, A calls func=on Reach(init, final, c.s( x ) ) where Reach(x, y, i) is true iff configura=on y of M can be reached from configura=on x in at most 2 i steps

15 Savitch s Theorem: NPSPACE = PSPACE Reach(x, y, i) can be implemented recursively, with recursion depth i The space per recursion level is propor=onal to the space used by M Reach(init, final, c.s( x ) ) uses space O(s( x ) 2

16 Periodic Graph Colouring Mo=va=on: schedule jobs at the same =me period each day; want to minimize processors Example: C D A B C D A Succinct representa=on: A +1 B C D

17 Periodic Graph Colouring Succinct graph is 3-colourable, sugges=ng that we need 3 processors (since two jobs in overlapping =me intervals cannot be scheduled on the same processor) But the infinite graph is actually 2-colourable, and so we can use just 2 processors! A +1 B C D A A A B C D B C D B C D

18 Periodic Graph Colouring A periodic graph G is an infinite undirected graph, specified by a triple (V,E,E ) G's nodes: V i where V i = {v i v in V}, for all i in Z G's edges: ( E i ) ( E i ) where E i = {{u i v i } {u,v} in E} and E i = { {u i v i+1 } (u,v) in E }

19 Periodic Graph Colouring Given a periodic graph G = (V,E,E ) and a posi=ve number k, is G k-colourable? A +1 B C D Exercise: suggest a nondeterminis=c algorithm for Periodic Graph Colouring that runs in polynomial space

20 Summary We ve seen representa=ve problems from new space bounded complexity classes PSPACE: TQBF, Periodic Graph Colouring NL: Directed Graph Reachability Savitch s Theorem: PSPACE = NPSPACE (and for space construc=ble s(n), SPACE(s(n)) SPACE(s(n) 2 ) ) We can leverage Savitch s Theorem to simplify proofs that some problems are in PSPACE

21 Summary Decidable TQBF EXP PSPACE = NPSPACE NP P NL DGR L PSPACEcomplete NLcomplete Addi=on

22 Next Class More on space bounded complexity classes Reading: Arora-Barak 4.3, 4.4