Dynamic Programming II

1. HKOI2009 Training (Advanced Group) Dynamic Programming II (Reference: Powerpoint of Dynamic Programming II, HKOI Training 2005, by Liu Chi Man, cx)

2. Review • Recurrence relation • Dynamic programming • State & Recurrence Formula • Optimal substructure • Overlapping subproblems

3. Outline • Dimension reduction (memory) • “Ugly” optimal value functions • DP on tree structures • Two-person games

4. Dimension reduction • Reduce the space complexity by one or more dimensions • “Rolling” array • Recall: Longest Common Subsequence (LCS) • Base conditions and recurrence relation: • Fi,0 = 0 for all i • F0,j = 0 for all j • Fi,j = Fi-1,j-1 + 1 (if A[i] = B[j]) max{ Fi-1,j , Fi,j-1 } (otherwise)

5. 0 0 0 0 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 0 1 0 0 2 0 3 0 4 0 Dimension Reduction • A: stxc, B: sicxtc 4 2 1 1 1 1 1 2 1 2 2 2 2 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 2 2 1 1 1 2 2 2 1 1 2 2 2 3

6. Dimension Reduction • We may discard old table entries if they are no longer needed • Instead of “rolling” the rows, we may “roll” the columns • Even less memory (52 entries) • Space complexity: (min{N, M}) • Drawback • Backtracking is difficult • That means we can get the number but not the sequence easily.

7. Packing 8 cross pieces onto a 10  6 grid (Simplified) Cannoneer Base • How many non-overlapping cross pieces can be put onto a HW grid? • W ≤ 10, H is arbitrary • A cross piece: • There may be patterns, but we just focus on a DP solution

8. ? k -2 k -1 k (Simplified) Cannoneer Base • We place the pieces from top to bottom • Phase k - putting all pieces centered on row k-1 • In phase k, we only need to consider the occupied squares in rows k-2 and k-1 Phase 3 Phase 4 Phase 5 Phase 6 Phase 7 Phase 8 Phase 9 Phase 10

9. (Simplified) Cannoneer Base • The optimal value function C is defined by: • C(k,S) = the max number of pieces after phase k, with rows k-1 and k giving the shape S • How to represent a shape? • In a shape, each column can be • Use 2, 1, 0 to represent these 3 cases • A shape is a W-digit base-3 integer • For example, the following shape is encoded as 010121(3) = 97(10)

10. (Simplified) Cannoneer Base • The recurrence relation is easy to construct • Max possible number of states = H  3W • That’s why W ≤ 10 • Cannoneer Base appeared in NOI2001 • Bugs Integrated, Inc. in CEOI2002 requires similar techniques

11. Dynamic Programming on Tree Structures • States may be (related to) nodes on a graph • Usually directed acyclic graphs • Topological order is the obvious order of recurrence evaluation • Trees are special graphs • A lot of graph DP problems are based on trees • Two major types: • Rooted tree DP • Unrooted tree DP

12. Rooted Tree Dynamic Programming • Base cases at the leaves • Recurrence at a node involves its child nodes only • Solution • Evaluate the recurrence relation from leaves (bottom) to the root (top) • Top-down implementations work well • Time complexity is often (N) where N is the number of nodes

13. Unrooted Tree Dynamic Programming • No explicit roots given • Two cases • The problem can be transformed to a rooted one • It can’t, so we try root every node • Case 2 increases the time complexity by a factor of N • Sometimes it is possible to root one node in O(N) time and each subsequent node in O(1) • overall O(N) time

14. Node Heights • Given a rooted tree T • The height of a node v in T is the maximum distance between v and a descendant of v • For example, all leaves have height = 0 • Find the heights of all nodes in T • Notations • C(v) = the set of children of v • p(v) = the parent of v

15. Node Heights • Optimal value function • H(v) = height of node v • Base conditions • H(u) = 0 for all leaves u • Recurrence • H(v) = max { H(x) | x C(v) } + 1 • Order of evaluation • All children must be evaluated before self • Post-order

16. Node Heights • Example A H(A) = 3 B C H(C) = 1 H(B) = 2 H(D) = 1 D E F G H H(E) = 0 H(F) = 0 H(G) = 0 H(H) = 0 I H(I) = 0

17. Node Heights • Time complexity analysis • Naively • There are N nodes • A node may have up to N-1 children • Overall time complexity = O(N2) • A better bound • The H-value of a v is at most used by one other node – p(v) • The total number of H-values inside the “max {}”s = N-1 • Overall time complexity = (N)

18. Treasure Hunt • N treasures are hidden at the N nodes of a tree (unrooted) • The treasure at node u has value v(u) • You may not take away two treasures joined by an edge, otherwise missiles will fly to you • Find the maximum value you can take away

19. Treasure Hunt • Let’s see if the problem can be transformed to a rooted one • We arbitrarily root a node, say r • How to formulate?

20. Treasure Hunt • Optimal value function • Z(u,b) = max value for the subtree rooted at u and it is b that the treasure at u is taken away • b = true or false • Base conditions • Z(x,false) = 0 and Z(x,true) = v(x) for all leaves x • Recurrence • Z(u,true) =  Z(c, false) + v(u) • Z(u,false) =  max { Z(c,false), Z(c,true) } • Answer = max { Z(r,false), Z(r,true) } c C(u) c C(u)

21. Treasure Hunt • Example (values shown in squares) false: 20 true: 27 7 false: 12 true: 3 false: 8 true: 6 2 6 false: 1 true: 9 9 3 5 1 2 false: 0 true: 3 false: 0 true: 5 false: 0 true: 1 false: 0 true: 2 false: 0 true: 1 1

22. Treasure Hunt • Our formulation does not exploit the properties of a tree root • Moreover the correctness of our formulation can be proven by optimal substructure • Thus the unrooted-to-rooted transformation is correct • Time complexity: (N)

23. Request Response Unrooted Tree DP – Basic Idea • In rooted tree DP, a node asks (request) for information from its children; and then provides (response) information to its parent

24. A Request Response B C D E Unrooted Tree DP – Basic Idea • In unrooted tree DP, a node also makes a request to its parent and sends response to its children • Imagine B is the root • A sends information about the “subtree” {A,C} to B

25. A A B C B C D E D E Unrooted Tree DP – Basic Idea • Similarly we can root C, D, E and get different request-response flows • These flows are very similar • The idea of unrooted tree DP is to root all nodes without resending all requests and responses every time

26. A B C D E Unrooted Tree DP – Basic Idea • Root A and do a complete flow • A knows about subtrees {B,D,E} and {C} • Now B sends a request to A • A sends a response to B telling what it knows about {A,C} • B already knows about {D}, {E} • Rooting of B completes

27. A B C D E Unrooted Tree DP – Basic Idea • Now let’s root D • D sends a request to B • B knows about {A,C}, {D}, and {E}; combining {A,C}, {E} and B itself, B knows about {B,A,C,E}, and sends a response to D • Rooting of D completes

28. A B C D E Unrooted Tree DP – Basic Idea • Rooting a new node requires only one request and one response if its parent already knows about all its subtrees (including the “imaginary” parent subtree) • Further questions: • Fast computation of {B,A,C,E} from {A,C} and {E}? (rooting of D) • Fast computation of {B,A,C,E,D} from {A,C}, {E}, {D}? (rooting of B)

29. Shortest Rooted Tree • Given an unrooted tree T, denote its rooted tree with root r by T(r) • Find a node v such that T(v) has the minimum height among all T(u), u T • The height of a tree = the height of its root • Solution • Just root every node and find the min height • We know how to find a height of a tree • Trivially this is (N2) • Now let’s use what we learnt

30. Shortest Rooted Tree • Since parents and children are unclear now, we use slightly different notations • N(v) = the set of neighbors of v • H(v, u) = height of the subtree rooted at v if u is treated as the parent of v • H(v, ) = height of the whole tree if v is root

31. Shortest Rooted Tree • Root A, complete flow • Height = 3 A H(A,) = 3 H(B,A) = 2 H(C,A) = 0 B C H(E,B) = 0 H(D,B) = 1 D E F H(F,D) = 0

32. Shortest Rooted Tree • Root B • Request: B asks A for H(A,B) • How can A give the answer in constant time? A H(A,) = 3 H(B,A) = 2 H(C,A) = 0 B C H(E,B) = 0 H(D,B) = 1 D E F H(F,D) = 0

33. Shortest Rooted Tree • Suppose now B asks A for H(A,B), how can A give the answer in constant time? • Two cases • B is the only largest subtree of A in T(A) • B is not the only largest subtree, or B is not a largest subtree A H(A,)=10 B C D E F G H I J K H(B,A)=7 H(D,A)=1 H(F,A)=9 H(H,A)=4 H(J,A)=3 H(C,A)=8 H(E,A)=2 H(G,A)=5 H(I,A)=6 H(K,A)=0

34. Shortest Rooted Tree • (1) B is the only largest subtree of A in T(A) • H(A,B) < H(A,) • H(A,B) depends on the second largest subtree • Trick: record the second largest subtree of A • (2) B is not the only largest subtree, or B is not a largest subtree • H(A,B) = H(A,)

35. Shortest Rooted Tree • To distinguish case (1) from case(2), we need to record the two largest subtrees of A • When? • When we evaluate H(A,) • Back to our example

36. Shortest Rooted Tree • Root B • Request: B asks A for H(A,B) • Response: 1 A H(A,) = 3 1st = B, 2nd = C H(A,B) = 1 H(B,A) = 2 H(C,A) = 0 B C 1st = D, 2nd = E A 1st = , 2nd =  H(E,B) = 0 H(D,B) = 1 D E 1st = F, 2nd =  1st = , 2nd =  F H(F,D) = 0 1st = , 2nd = 

37. Shortest Rooted Tree • Root B • H(B,) = 2 can be calculated in constant time A H(A,) = 3 1st = B, 2nd = C H(A,B) = 1 H(B,) = 2 H(B,A) = 2 H(C,A) = 0 B C 1st = D, 2nd = E A 1st = , 2nd =  H(E,B) = 0 H(D,B) = 1 D E 1st = F, 2nd =  1st = , 2nd =  F H(F,D) = 0 1st = , 2nd = 

38. Shortest Rooted Tree • Root D • Request: D asks B for H(B,D) • Response: 2 • H(D,) = 3 A H(A,) = 3 1st = B, 2nd = C H(A,B) = 1 H(B,D) = 2 H(B,) = 2 H(B,A) = 2 H(C,A) = 0 B C 1st = D, 2nd = E A 1st = , 2nd =  H(D,) = 3 H(E,B) = 0 H(D,B) = 1 D E 1st = F, 2nd =  1st = , 2nd =  B F F H(F,D) = 0 1st = , 2nd = 

39. Shortest Rooted Tree • Root F, E, and C in the same fashion • In general, root the nodes in preorder • Time complexity analysis • Root A – (N) • Root each subsequent nodes – O(1) • Overall - (N) • The O(1) is crucial for the linearity of our algorithm • If rooting of a new node cannot be done fast, unrooted tree DP may not improve running time

40. Two-person Games • Often appear in competitions as interactive tasks • Playing against the judge • Most of them can be solved by the Minimax method

41. o o o o x x o x o x x o x o o o Game Tree • A (finite or infinite) rooted tree showing the movements of a game play … … … … … … … …

42. 2 9 4 5 6 6 8 1 7 6 Game Tree • This is a game • The boxes at the bottom show your gain (your opponent’s loss) • Your opponent is clever • How should you play to maximize your gain? WHY? Your turn Her turn End of game

43. Minimax • You assume that your opponent always try to minimize her loss (minimize your gain) • So your opponent always takes the move that minimize your gain • Of course, you always take the move that maximize your gain 7 Your turn Her turn 4 6 6 7 9 4 5 6 8 7 2 9 4 5 6 6 8 1 7 6

44. A A B C D C D D B 2 1 2 2 1 2 1 2 D 1 Minimax • Efficient? • Only if the tree is small • In fact the game tree may in fact be an expanded version of a directed acyclic graph • Overlapping subproblems  memo(r)ization

45. Past Problems • IOI • 2001 Ioiwari (game), Score (game), Twofive (ugly) • 2005 Rivers(tree) • NOI • 2001 炮兵陣地(ugly), 2002 貪吃的九頭龍 (tree), 2003 逃學的小孩(tree), 2005 聰聰與可可 • IOI/NOITFT • 2004 A Bomb Too Far (tree) • CEOI • 2002 Bugs (ugly), 2003 Pearl (game) • Balkan OI • 2003 Tribe (tree) • Baltic OI • 2003 Gems (tree) • On HKOI Judge • 1074 Christmas Tree