Dynamic Programming Greedy. Build up a solution incrementally, - - PowerPoint PPT Presentation

Dynamic Programming

Introduction, Weighted Interval Scheduling Tyler Moore

CS 2123, The University of Tulsa

Some slides created by or adapted from Dr. Kevin Wayne. For more information see http://www.cs.princeton.edu/~wayne/kleinberg-tardos. Some code reused from Python Algorithms by Magnus Lie Hetland.

Algorithmic paradigms

• Greedy. Build up a solution incrementally, myopically optimizing

some local criterion. Divide-and-conquer. Break up a problem into independent subproblems, solve each subproblem, and combine solution to subproblems to form solution to original problem. Dynamic programming. Break up a problem into a series of overlapping subproblems, and build up solutions to larger and larger subproblems.

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fancy name for caching away intermediate results in a table for later reuse

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Dynamic programming history

• Bellman. Pioneered the systematic study of dynamic programming in 1950s.

Etymology.

・Dynamic programming = planning over time. ・Secretary of Defense was hostile to mathematical research. ・Bellman sought an impressive name to avoid confrontation.

3 THE THEORY OF DYNAMIC PROGRAMMING

RICHARD BELLMAN

• 1. Introduction. Before turning to a discussion of some representa-

tive problems which will permit us to exhibit various mathematical features of the theory, let us present a brief survey of the funda- mental concepts, hopes, and aspirations of dynamic programming. To begin with, the theory was created to treat the mathematical problems arising from the study of various multi-stage decision processes, which may roughly be described in the following way: We have a physical system whose state at any time / is determined by a set of quantities which we call state parameters, or state variables. At certain times, which may be prescribed in advance, or which may be determined by the process itself, we are called upon to make de- cisions which will affect the state of the system. These decisions are equivalent to transformations of the state variables, the choice of a decision being identical with the choice of a transformation. The out- come of the preceding decisions is to be used to guide the choice of future ones, with the purpose of the whole process that of maximizing some function of the parameters describing the final state. Examples of processes fitting this loose description are furnished by virtually every phase of modern life, from the planning of indus- trial production lines to the scheduling of patients at a medical clinic ; from the determination of long-term investment programs for universities to the determination of a replacement policy for ma- chinery in factories; from the programming of training policies for skilled and unskilled labor to the choice of optimal purchasing and in- ventory policies for department stores and military establishments. It is abundantly clear from the very brief description of possible applications that the problems arising from the study of these processes are problems of the future as well as of the immediate present. Turning to a more precise discussion, let us introduce a small amount of terminology. A sequence of decisions will be called a policy, and a policy which is most advantageous according to some preassigned criterion will be called an optimal policy. The classical approach to the mathematical problems arising from the processes described above is to consider the set of all possible An address delivered before the Summer Meeting of the Society in Laramie on September 3, 1953 by invitation of the Committee to Select Hour Speakers for An- nual and Summer meetings; received by the editors August 27,1954. 503

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Dynamic programming applications

Areas.

・Bioinformatics. ・Control theory. ・Information theory. ・Operations research. ・Computer science: theory, graphics, AI, compilers, systems, …. ・...

Some famous dynamic programming algorithms.

・Unix diff for comparing two files. ・Viterbi for hidden Markov models. ・De Boor for evaluating spline curves. ・Smith-Waterman for genetic sequence alignment. ・Bellman-Ford for shortest path routing in networks. ・Cocke-Kasami-Younger for parsing context-free grammars. ・...

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Recurrence relations

Recall that recurrence relations are equations defined in terms of

• themselves. They are useful because many natural functions and

recursive functions can easily expressed as recurrences Recurrence Solution Example application T(n) = T(n/2) + 1 Θ(lg n) Binary Search T(n) = T(n/2) + n Θ(n) Randomized Quickselect (avg. case) T(n) = 2T(n/2) + 1 Θ(n) Tree traversal T(n) = 2T(n/2) + n Θ(n lg n) Mergesort T(n) = T(n − 1) + 1 Θ(n) Processing a sequence T(n) = T(n − 1) + n Θ(n2) Handshake problem T(n) = 2T(n − 1) + 1 Θ(2n) Towers of Hanoi T(n) = 2T(n − 1) + n Θ(2n) T(n) = nT(n − 1) Θ(n!)

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Computing Fibonacci numbers

Fibonacci sequence can be defined using the following recurrence: Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1 F2 = F1 + F0 = 1 + 0 = 1 F3 = F2 + F1 = 1 + 1 = 2 F4 = F3 + F2 = 2 + 1 = 3 F5 = F4 + F3 = 3 + 2 = 5 F6 = F5 + F4 = 5 + 3 = 8

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Computing Fibonacci numbers with recursion

def f i b ( i ) : i f i < 2: return i return f i b ( i −1) + f i b ( i −2)

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Recursion Tree for Fibonacci function

F(6) F(5) F(4) F(3) F(2) F(1) F(0) F(1) F(2) F(1) F(0) F(3) F(2) F(1) F(0) F(1) F(4) F(3) F(2) F(1) F(0) F(1) F(2) F(1) F(0)

We know that T(n) = 2T(n − 1) + 1 = Θ(2n) It turns out that T(n) = T(n − 1) + T(n − 2) ≈ Θ(1.6n) Since our recursion tree has 0 and 1 as leaves, computing Fn requires ≈ 1.6n recursive function calls!

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Computing Fibonacci numbers with memoization (manual)

def fib memo ( i ) : mem = {} #d i c t

• f

cached v a l u e s def f i b ( x ) : i f x < 2: return x #check i f a l r e a d y computed i f x in mem: return mem[ x ] #only i f not a l r e a d y computed mem[ x ] = f i b ( x−1) + f i b ( x−2) return mem[ x ] f i b ( i ) return mem[ i ]

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Recursion tree for Fibonacci function with memoization

F(6) F(5) F(4) F(3) F(2) F(1) F(0) F(1) F(2) F(1) F(0) F(3) F(2) F(1) F(0) F(1) F(4) F(3) F(2) F(1) F(0) F(1) F(2) F(1) F(0)

Black nodes: no longer computed due to memoization Caching reduced the # of operations from exponential to linear time!

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Computing Fibonacci numbers with memoization (automatic)

> > > @memo . . . def f i b ( i ) : . . . i f i < 2: return i . . . return f i b ( i −1) + f i b ( i −2) . . . > > > f i b (100) 354224848179261915075L What sort of magic is going on here?

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Code for the memo wrapper

from f u n c t o o l s import wraps def memo( func ) : cache = {} # Stored subproblem s o l u t i o n s @wraps ( func ) # Make wrap look l i k e func def wrap (∗ args ) : # The memoized wrapper i f args not in cache : # Not a l r e a d y computed? cache [ args ] = func (∗ args ) # Compute & cache the s o l u return cache [ args ]# Return the cached s o l u t i o n return wrap # Return the wrapper Example of Python’s capability as a functional language Provides cache functionality for recursive functions in general The memo function takes a function as input, then “wraps the function with the added functionality The @wraps statement makes the memo function a decorator

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Discussion of dynamic memoization

Even if the code is a bit of a mystery, don’t worry, you can still use it by including the code on the last slide with yours, then making the first line before your function definition decorated by ‘@memo’ If you don’t have access to a programming language supporting dynamic memoization, you can either do it manually or turn to dynamic programming Dynamic programming converts recursive code to an iterative version that executes efficiently

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Computing Fibonacci numbers with dynamic programming

def f i b i t e r ( i ) : i f i < 2: return i #s t o r e the sequence in a l i s t mem=[0 ,1] for j in range (2 , i +1): #i n c r e m e n t a l l y b u i l d the sequence

• mem. append (mem[ j −1]+mem[ j −2])

return mem[ −1]

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Avoiding recomputation by storing partial results

The trick to dynamic programming is to see that the naive recursive algorithm repeatedly computes the same subproblems over and over and over again. If so, storing the answers to them in a table instead

• f recomputing can lead to an efficient algorithm.

Thus we must first hunt for a correct recursive algorithm – later we can worry about speeding it up by using a results matrix.

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SECTION 6.1-6.2

• 6. DYNAMIC PROGRAMMING I
• weighted interval scheduling
• segmented least squares
• knapsack problem
• RNA secondary structure

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Weighted interval scheduling

Weighted interval scheduling problem.

・Job j starts at sj, finishes at fj, and has weight or value vj. ・Two jobs compatible if they don't overlap. ・Goal: find maximum weight subset of mutually compatible jobs.

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time

f g h e a b c d

1 2 3 4 5 6 7 8 9 10 11

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Earliest-finish-time first algorithm

Earliest finish-time first.

・Consider jobs in ascending order of finish time. ・Add job to subset if it is compatible with previously chosen jobs.

• Recall. Greedy algorithm is correct if all weights are 1.
• Observation. Greedy algorithm fails spectacularly for weighted version.

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weight = 999 weight = 1 time 1 2 3 4 5 6 7 8 9 10 11

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• Notation. Label jobs by finishing time: f1 ≤ f2 ≤ . . . ≤ fn .
• Def. p ( j ) = largest index i < j such that job i is compatible with j.
• Ex. p(8) = 5, p(7) = 3, p(2) = 0.

Weighted interval scheduling

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time 1 2 3 4 5 6 7 8 9 10 11

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Dynamic programming: binary choice

• Notation. OPT(j) = value of optimal solution to the problem consisting of

job requests 1, 2, ..., j. Case 1. OPT selects job j.

・Collect profit vj. ・Can't use incompatible jobs { p(j) + 1, p(j) + 2, ..., j – 1 }. ・Must include optimal solution to problem consisting of remaining

compatible jobs 1, 2, ..., p(j). Case 2. OPT does not select job j.

・Must include optimal solution to problem consisting of remaining

compatible jobs 1, 2, ..., j – 1. OPT( j) = if j= 0 max v j + OPT( p( j)), OPT( j −1)

{ }

• therwise

⎧ ⎨ ⎩

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• ptimal substructure property

(proof via exchange argument)

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Weighted interval scheduling: brute force

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Input: n, s[1..n], f[1..n], v[1..n] Sort jobs by finish time so that f[1] ≤ f[2] ≤ … ≤ f[n]. Compute p[1], p[2], …, p[n]. Compute-Opt(j) if j = 0 return 0. else return max(v[j] + Compute-Opt(p[j], Compute-Opt(j–1))).

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Weighted interval scheduling: brute force

• Observation. Recursive algorithm fails spectacularly because of redundant

subproblems ⇒ exponential algorithms.

• Ex. Number of recursive calls for family of "layered" instances grows like

Fibonacci sequence.

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3 4 5 1 2 p(1) = 0, p(j) = j-2

5 4 3 3 2 2 1 1 1 1 1 1

recursion tree

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• Memoization. Cache results of each subproblem; lookup as needed.

global array

Weighted interval scheduling: memoization

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Input: n, s[1..n], f[1..n], v[1..n] Sort jobs by finish time so that f[1] ≤ f[2] ≤ … ≤ f[n]. Compute p[1], p[2], …, p[n]. for j = 1 to n M[j] ← empty. M[0] ← 0. M-Compute-Opt(j) if M[j] is empty M[j] ← max(v[j] + M-Compute-Opt(p[j]), M-Compute-Opt(j – 1)). return M[j].

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Weighted interval scheduling: running time

• Claim. Memoized version of algorithm takes O(n log n) time.

・Sort by finish time: O(n log n). ・Computing p(⋅) : O(n log n) via sorting by start time. ・M-COMPUTE-OPT(j): each invocation takes O(1) time and either

• (i) returns an existing value M[j]
• (ii) fills in one new entry M[j] and makes two recursive calls

・Progress measure Φ = # nonempty entries of M[].

• initially Φ = 0, throughout Φ ≤ n.
• (ii) increases Φ by 1 ⇒ at most 2n recursive calls.

・Overall running time of M-COMPUTE-OPT(n) is O(n). ▪

• Remark. O(n) if jobs are presorted by start and finish times.

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Weighted interval scheduling: bottom-up

Bottom-up dynamic programming. Unwind recursion.

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BOTTOM-UP (n, s1, …, sn , f1, …, fn , v1, …, vn)

Sort jobs by finish time so that f1 ≤ f2 ≤ … ≤ fn. Compute p(1), p(2), …, p(n). M [0] ← 0. FOR j = 1 TO n M [ j ] ← max { vj + M [ p(j) ], M [ j – 1] }.

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Weighted Interval Scheduling Example

Index 1 2 3 4 5 6 v1 = 2 v2 = 4 v3 = 4 v4 = 7 v5 = 2 v6 = 1

j vj p(j) vj + M[p(j)] M[j − 1] M[j] 1 2 2 + 0 2 2 4 4 + 0 2 4 3 4 1 4 + 2 4 6 4 7 7 + 0 6 7 5 2 3 2 + 6 7 8 6 1 3 1 + 6 8 8

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Weighted Interval Scheduling Exercise

Index 1 2 3 4 5 v1 = 2 v2 = 3 v3 = 6 v4 = 1 v5 = 9

j vj p(j) vj + M[p(j)] M[j − 1] M[j] 1 2 2 3 3 6 4 1 5 9

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Weighted interval scheduling: finding a solution

• Q. DP algorithm computes optimal value. How to find solution itself?
• A. Make a second pass.
• Analysis. # of recursive calls ≤ n ⇒ O(n).

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Find-Solution(j) if j = 0 return ∅. else if (v[j] + M[p[j]] > M[j–1]) return { j } ∪ Find-Solution(p[j]). else return Find-Solution(j–1).

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