CS141 BB: DynamicProgramming

Dynamic Programming references

CountingPathsByDP
FibonacciByDP
NChooseKByDP
SubsetSumByDP
MinWtTriangulationByDP
KnapsackByDP
TransitiveClosureByDP
ShortestPathsByDP
GoodrichAndTomassia sections 5.3.3 (0/1 knapsack), 6.4.2 (transitive closure), 7.1, 7.2.1

summary

problem subproblem for each recurrence base case final answer run time

count # s-> t paths P[v] = #paths from s to v vertex in graph P[v] = ∑_{w: (u,w)∈ E} P[w] P[s] = 1 P[t] O(#vertices+#edges)

n choose k C[i,j] =
#size-j subsets of {1,2,...,i} i=0,1,..,n; j=0,1,...,i C[n,k] =
C[n-1,k] + C[n-1,k-1] C[n,0] = 1
C[0,i] = 0 (i > 0) C[n,k] O(n*k)

∃ subset of S[1..n]
summing to g? ES[i,h] = true
if subset of {1,2,..,i}
sums to h i=0,1,..,n; h=0,1,...,g ES[i,h] =
ES[i-1,h] or ES[i-1,h-S[i]] ES[0,0] = true ES[n,g] O(n*g)

min wt triangulation
of {1,2,...,n} MWT[i,j] = min wt
triangulation of {i,i+1,..,j} 1 ≤ i < j ≤ n MWT[i,j] = min
{cost[i,j] + MWT[i,k] + MWT[k,j] :
k=i+1,i+2,...,j-1 } MWT[i,i+1] = cost[i,i+1] MWT[1,n] O(n³)

longest ascending
subsequence of {S[1],S[2],...,S[n]} LAS[i] = longest
ascending subsequence
of {S[1],..S[i]} ending at S[i] i=1,2,...,n LAS[i] = max{ 1, max{ 1 + LAS[j] : j=1,2,..,i-1 and S[j] < S[i] } - max {LAS[i] : i=1,2,...,n} O(n²)

principles

1. ask what subproblems you would need to solve in order to solve given problem

2. make sure the number of distinct subproblems is not too large

3. formulate a recurrence between subproblems

4. use the recurrence to design a recursive algorithm that caches, or an iterative algorithm

problem	subproblem	for each	recurrence	base case	final answer	run time
count # s-> t paths	P[v] = #paths from s to v	vertex in graph	P[v] = ∑_{w: (u,w)∈ E} P[w]	P[s] = 1	P[t]	O(#vertices+#edges)
n choose k	C[i,j] = #size-j subsets of {1,2,...,i}	i=0,1,..,n; j=0,1,...,i	C[n,k] = C[n-1,k] + C[n-1,k-1]	C[n,0] = 1 C[0,i] = 0 (i > 0)	C[n,k]	O(n*k)
∃ subset of S[1..n] summing to g?	ES[i,h] = true if subset of {1,2,..,i} sums to h	i=0,1,..,n; h=0,1,...,g	ES[i,h] = ES[i-1,h] or ES[i-1,h-S[i]]	ES[0,0] = true	ES[n,g]	O(n*g)
min wt triangulation of {1,2,...,n}	MWT[i,j] = min wt triangulation of {i,i+1,..,j}	1 ≤ i < j ≤ n	MWT[i,j] = min {cost[i,j] + MWT[i,k] + MWT[k,j] : k=i+1,i+2,...,j-1 }	MWT[i,i+1] = cost[i,i+1]	MWT[1,n]	O(n³)
longest ascending subsequence of {S[1],S[2],...,S[n]}	LAS[i] = longest ascending subsequence of {S[1],..S[i]} ending at S[i]	i=1,2,...,n	LAS[i] = max{ 1, max{ 1 + LAS[j] : j=1,2,..,i-1 and S[j] < S[i] }	-	max {LAS[i] : i=1,2,...,n}	O(n²)