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Chapter 5
Dynamic Programming
충북대학교 알고리즘연구실
Dynamic Programming
• Used when the solution of a problem is a result of a sequence of decisions
• Example
– Knapsack
– Shortest path
– Optimal merge patterns
5.3 All pairs shortest paths
• G(V,E) : A directed graph with n vertices
• Ak(i,j) : Length of shortest path from i to j going through no vertex of index greater than k
• A(i,j) = min{min{Ak-1(i,k) + Ak-1(k,j)},cost(i,j)}
• Ak (i,j) = Ak-1(i,k) + Ak-1(k,j)
Ak (i,j) = min{Ak-1(i,j),Ak-1(i,k) + Ak-1(k,j)},k>=1
C(i,j) = 0, 1<=i<=n
C(i,j) Cost of edge if (i,j) E
if i j and (i,j) E ∞∈
∉≠
∴
5.3 All pairs shortest paths
• Figure 5.5 Graph with negative cycle
1 2 3 1 1
-2
∞∞−
∞
0102
10
5.3 All pairs shortest paths
• Program 5.3 Function to compute lengths of shortest paths
5.3 All pairs shortest paths
• Figure 5.6 Directed Graph and associated matrices
5.4 Single-source shortest paths: General weights
• dist[u] = cost[v][u]
• distl[u] = length of a shortest path from the source vertex V to vertex U under the constraint that the shortest path contains at most L edges
• 순환관계 – distk[u] = min(distk-1[u],min(distk-1[i] + cost[i][u]))
,2<=k<=n-1
5.4 Single-source shortest paths: General weights
• Figure 5.10 Shortest paths with negative edge lengths
5.4 Single-source shortest paths: General weights
• Program 5.4 Bellman and Ford algorithm to compute shortest paths
5.5 Optimal binary search trees
• Definition – A binary search tree T • All identifies in the Tleft < Troot
• All identifies in the Tright < Troot
• The left and right subtres of T are also BST
• 가정 – a1 < a2 < … < an
– Ti,j : OBST for ai+1,…,aj
– Ci,j : cost for Ti,j
– Ri,j : root of Ti,j
– Weight of Ti,j : Wi,j = Qi + ∑+=
+j
ikkk PQ
1)(
5.5 Optimal binary search trees
• P(i) : probability Search(a(i))
• Q(i) : probability search (a(i) < x < a(i+1))
• : Probability of an unsuccessful search
∑≤≤ ni
iQ0
)(
1)()(01
=+∴ ∑∑≤≤≤≤ nini
iQiPInternal node
External node
∑∑≤≤≤≤
−+ni
ini
i EleveliqaleveliP01
)1)((*)()(*)( (5.9)
Optimal Binary Search Tree
Cost of the search Tree
p(k)+cost(l)+cost(r)+w(0,k-1)+w(k,n)
c(0, n) = min{c(0, k-1) + c(k, n) +p(k)
+w(0, k-1)+w(k, n)}
c(i, j) = min{c(i, k-1), c(k, j)+p(k) +
w(i,k-1)+w(k,j)}
C(i,j) = min{c(i,k-1)+c(k,j)} + w(i,j)
5.5 Optimal binary search trees
• Figure 5.12 Two possible binary search trees
5.5 Optimal binary search trees
• Figure 5.13 Binary search trees of Figure 5.12 with external nodes added
5.5 Optimal binary search trees
5.5 Optimal binary search trees
5.5 Optimal binary search trees
• Figure 5.16 Computation of c(0,4), w(0,4), and r(0,4)
5.8 Reliability design
• Solve a problem with a multiplicative optimization function
• Several devices are connected in series
• ri be the reliability of device Di
• Reliability of the entire system
• Duplicate : multiple copies of the same device type are connected in parallel use switching circuits
∏ ir
5.8 Reliability design
Figure 5.19 n devices Di, 1<=i<=n, connected in series
Figure 5.20 Multiple devices connected in parallel in each stage
Multiple copies
stage in contain mi copies of Di
P(all mi malfunction) = (1-ri)mi
Reliability of stage i =1-(1-ri)mi
5.8 Reliability design
• Maximum allowable cost of the system Maximize
Subject to
Mi >=1 and integer, 1<=i<=n
• Assume ci>0 ui =
∏ ≤≤ ni ii m1
)(φ
cmcni
ii ≤∑≤≤1
−+ ∑ i
n
ji cccc /)(1
5.9 The traveling salesperson problem
• 우체부 : n개의 틀린 장소에서 mail pickup
– n+1 vertex graph
– Edge <i,j> distance from i to j
– Tour of minimum cost
• Permutation problem
– n! different permutation of n object while there are 2n different subset of n object
n! > O(2n)
5.9 The traveling salesperson problem
• Tour : simple path that starts and ends at vertex 1
• Every tour : edge<1,k> for some k v-{1} each <k,1>
• Optimal tour : path(k,1) – Shortest k to 1 path all the vertices in V-{1,k}
• Let g(i, S) be the length of a shortest path starting at vertex i, going through all vertices in S and terminating at vertex 1
})},1{,({min})1{,1( 12kVkgcVg knk
−+=−≤≤
})}{,({min),( jSjgcSig ijSj−+=
∈
∈
5.9 The traveling salesperson problem
• Figure 5.21 Directed graph and edge length matrix c
5.9 The traveling salesperson problem
• Thus g(2, ) = c21 = 5, g(3, ) = c31 = 6, and g(4, ) = c41 = 8. We obtain
g(2,{3}) = c23 + g(3, ) = 15 g(2,{4}) = 18
g(3,{2}) = 18 g(3,{4}) = 20
g(4,{2}) = 13 g(4,{3}) = 15
g(2,{3,4}) = min{c23+g(3,{4}),c24+g(4,{3})} = 25
g(3,{2,4}) = min{c32+g(2,{4}),c34+g(4,{2})} = 25
g(4,{2,3}) = min{c42+g(2,{3}),c43+g(3,{2})} = 23
g(1,{2,3,4}) = min{c12+g(2,{3,4}),c13+g(3,{2,4}),c14+g(4,{2,3})}
= min(35,40,43}
= 35
φ
φφ φ
5.9 The traveling salesperson problem
• Let N be the number of g(i,s), that have to be computed before g(1,V-{1}) i, computed for each value of |s|
• n-1 choices of i
• The number of distinct sets of S of size k not including 1 and i
∃
i
−k
n 2
22
02)1()
2)(1( −
−
=
−=−
−=∴ ∑ nn
kn
kn
nN
)2(lg 2 nnorithma θ=
)2()( nnOneedspace =
