Algorithm Design Methods

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Algorithm Design Methods

• Spring 2007
• CSE, POSTECH

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Algorithm Design Methods

• Greedy method
• Divide and conquer
• Dynamic programming
• Backtracking
• Branch and bound

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Some Methods Not Covered

• Linear Programming
• Integer programming
• Simulated annealing
• Neural networks
• Genetic algorithms
• Tabu search

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Optimization Problem

• A problem in which some function(called the
  optimization/objective function)is to be
  optimized (usually minimized or maximized)
• It is subject to some constraints.

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Machine Scheduling

• Find a schedule that minimizes the finish time.
• optimization function finish time
• constraints
• Each job is scheduled continuously on a single
  machine for an amount of time equal to its
  processing requirement.
• No machine processes more than one job at a time.

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Bin Packing

• Pack items into bins using fewest number of bins.
• optimization function number of bins
• constraints
• Each item is packed into a single bin.
• The capacity of no bin is exceeded.

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Min Cost Spanning Tree

• Find a spanning tree that has minimum cost.
• optimization function sum of edge costs
• constraints
• Must select n-1 edges of the given n vertex
  graph.
• The selected edges must form a tree.

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Feasible and Optimal Solutions

• A feasible solution is a solution that satisfies
  the constraints.
• An optimal solution is a feasible solution that
  optimizes the objective/optimization function.

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Greedy Method

• Solve problem by making a sequence of decisions.
• Decisions are made one by one in some order.
• Each decision is made using a greedy criterion.
• A decision, once made, is (usually) not changed
  later.

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Machine Scheduling

• LPT Scheduling.
• Schedule jobs one by one in decreasing order of
  processing time (i.e., longest processing time
  first).
• Each job is scheduled on the machine on which it
  finishes earliest.
• Scheduling decisions are made serially using a
  greedy criterion (minimize finish time of this
  job).
• LPT scheduling is an application of the greedy
  method.

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LPT Schedule

• LPT rule does not guarantee minimum finish time
  schedules.
• (LPT Finish Time)/(Minimum Finish Time) lt 4/3
  1/(3m)
• where m is number of machines.
• Minimum finish time scheduling is NP-hard.
• In this case, the greedy method does not work.
• Greedy method does, however, give us a good
  heuristic for machine scheduling.

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Container Loading Problem

• Ship has capacity c.
• m containers are available for loading.
• Weight of container i is wi.
• Each weight is a positive number.
• Sum of container weight lt c.
• Load as many containers as possible without
  sinking the ship.

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Greedy Solution

• Load containers in increasing order of weight
  until we get to a container that does not fit.
• Does this greedy algorithm always load the
  maximum number of containers?
• Yes. May be proved using a proof by
  induction.(see Theorem 13.1, p. 624 of text.)

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Container Loading with 2 Ships

• Can all containers be loaded into 2 ships whose
  capacity is c each?
• Same as bin packing with 2 bins(Are 2 bins
  sufficient for all items?)
• Same as machine scheduling with 2 machines(Can
  all jobs be completed by 2 machines in c time
  units?)
• NP-hard

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0/1 Knapsack Problem

• Hiker wishes to take n items on a trip.
• The weight of item i is wi.
• The knapsack has a weight capacity c.
• If sum of items weights lt c,all n items can be
  carried in the knapsack.
• If sum of item weights gt c,some items must be
  left behind.
• Which items should be taken out?

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0/1 Knapsack Problem

• Hiker assigns a profit/value pi to item i.
• All weights and profits are positive numbers.
• Hiker wants to select a subset of n items to
  take.
• The weight of the subset should not exceed the
  capacity of the knapsack. (constraint)
• Cannot select a fraction of an item. (constraint)
• The profit/value of the subset is the sum of the
  profits of the selected items. (optimization
  function)
• The profit/value of the selected subset should be
  maximum. (optimization criterion)

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0/1 Knapsack Problem

• Let xi1 when item i is selected andlet xi0
  when item i is not selected.
• maximize Sigma(i1n) pixi
• subject to Sigma(i1n) wixi lt c
• See the formula and constraints on page 625

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Greedy Attempt 1

• Be greedy on capacity utilization(select items
  in increasing order of weights).
• n 2, c 7
• w 3, 6
• p 2, 10
• Only 1 item is selected, x 1, 0.Profit/value
  of selection is 2.It is not the best selection.

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Greedy Attempt 2

• Be greedy on profit earned(select items in
  decreasing order of profit).
• n 3, c 7
• w 7, 3, 2
• p 10, 8, 6
• Only 1 item is selected , x 1, 0,
  0.Profit/value of selection is 10.It is not
  the best selection.

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Greedy Attempt 3

• Be greedy on profit density (p/w)(select items
  in decreasing order of profit density).
• n 2, c 7
• w 1, 7
• p 10, 20
• Only 1 item is selected, x 1, 0.Profit/value
  of selection is 10.It is not the best selection.

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Greedy Attempt 4

• Select a subset with lt k items.
• If the weight of this subset is gt c,discard the
  subset.
• If the subset weight is lt c,fill as much of the
  remaining capacity as possible by being greedy on
  profit density.
• Try all subsets with lt k items andselect the
  one that yields maximum profit.

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Chapter 14 Divide and Conquer

• A large problem is solved as follows
• Divide the large problem into smaller problems.
• Solve the smaller problems somehow.
• Combine the results of the smaller problemsto
  obtain the result for the original large problem.
• A small problem is solved in some other way.

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Small and Large Problem

• Small problem
• Sort a list that has n lt 10 elements.
• Find the minimum of n lt 2 elements.
• Large problem
• Sort a list that has n gt 10 elements.
• Find the minimum of n gt 2 elements.

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Solving a Small Problem

• A small problem is solvedusing some
  direct/simple strategy.
• Sort a list that has n lt 10 elements.Use
  insertion, bubble, or selection sort.
• Find the minimum of n lt 2 elements.When n 0,
  there is no minimum element.When n 1, the
  single element is the minimum.When n 2,
  compare the two elements and determine which is
  smaller.

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Sort a Large List

• Sort a list that has n gt 10 elements.
• Sort 15 elements by dividing them into 2 smaller
  lists.One list has 7 elements and the other has
  8 elements.
• Sort these two lists using the method for small
  lists.
• Merge the two sorted lists into a single sorted
  list.

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Find the Min of a Large List

• Find the minimum of 20 elements.
• Divide into two groups of 10 elements each.
• Find the minimum element in each group somehow.
• Compare the minimums of each group to determine
  the overall minimum.

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Recursion In Divide and Conquer

• Often the smaller problems that result from the
  divide step are instances of the original
  problem(true for our sort and min problems). In
  this case,
• If the new problem is a smaller problem,it is
  solved using the method for small problems.
• If the new problem is a large instance, it is
  solvedusing the divide-and-conquer method
  recursively.
• Generally, performance is best when the smaller
  problems that result from the divide step are of
  approximately the same size.

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Recursive Find Min

• Find the minimum of 20 elements.
• Divide into two groups of 10 elements each.
• Find the minimum element in each group
  recursively.The recursion terminates when the
  number of elementsis lt 2. At this time the
  minimum is found using the method for small
  problems.
• Compare the minimums of each group to determine
  the overall minimum.

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Merge Sort another Divide Conquer Example

• Sort the first half of the array using merge
  sort.
• Sort the second half of the array using merge
  sort.
• Merge the first half of the array with the second
  half.

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Merge Sort Algorithm

• Merge is an operation that combines two sorted
  arrays.
• Assume the result is to be placed in a separate
  array called result (already allocated).
• The two given arrays are called front and back.
• front and back are in increasing order.
• For the complexity analysis,the size of the
  input, n, is the sum nfront nback.

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Merge Sort Algorithm

• For each array keep track of the current
  position.
• REPEAT until all the elements of one of the given
  arrays have been copied into result
• Compare the current elements of front and back.
• Copy the smaller into the current position of
  result (break the ties however you like).
• Increment the current position of result and the
  array that was copied from.
• Copy all the remaining elements of the other
  given array into result.

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Merge Sort Algorithm - Complexity

• Every element in front and back is copied exactly
  once. Each copy is two accesses, so the total
  number of accessing due to copying is 2n.
• The number of comparisons could beas small as
  min(nfront, nback) or as large as n-1.Each
  comparison is two accesses.

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Merge Sort Algorithm - Complexity

• In the worst casethe total number of accesses
  is2n 2(n-1) O(n).
• In the best casethe total number of accesses
  is2n 2min(nfront,nback) O(n).
• The average case is between the worst and best
  case and is therefore also O(n).

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Merge Sort Algorithm

• Split anArray into two non-empty parts anyway you
  like. For example,front the first n/2 elements
  in anArrayback the remaining elements in
  anArray
• Sort front and back by recursively calling
  MergeSort.
• Now you have two sorted arrays containing all the
  elements from the original array.Use merge to
  combine them, put the result in anArray.

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MergeSort Call Graph (n7)

• Each box represents one invocation of MergeSort.
• How many levels are there in generalif the array
  is divided in half each time?

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MergeSort Call Graph (general)

• Suppose n 2k. How many levels?
• How many boxes on level j?
• What values is in each box at level j?

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Quick Sort

• Quicksort can be seen as a variation of mergesort
  in which front and back are defined in a
  different way.

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Quicksort Algorithm

• Partition anArray into two non-empty parts.
• Pick any value in the array, pivot.
• small the elements in anArray lt pivot
• large the elements in anArray gt pivot
• Place pivot in either part,so as to make sure
  neither part is empty.
• Sort small and large by recursively calling
  QuickSort.
• How would you merge the two arrays?
• You could use merge to combine them, but because
  the elements in small are smaller than elements
  in large, simply concatenate small and large, and
  put the result into anArray.

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Quicksort Complexity Analysis

• Like mergesort,a single invocation of quicksort
  on an array of size phas complexity O(p)
• p comparisons 2p accesses
• 2p moves (copying) 4p accesses
• Best case every pivot chosen by quicksort
  partitions the array into equal-sized parts. In
  this case quicksort is the same big-O complexity
  as mergesort O(n log n)

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Quicksort Complexity Analysis

• What would be the worst case scenario?
• Worst case the pivot chosen is the largest or
  smallest value in the array. Partition creates
  one part of size 1 (containing only the pivot),
  the other of size p-1.

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Quicksort Complexity Analysis

• Worst case
• There are n-1 invocations of quicksort (not
  counting base cases) with arrays of sizep n,
  n-1, n-2, , 2
• Since each of these does O(p),the total number
  of accesses isO(n) O(n-1) O(1) O(n2)
• Ironically the worst case occurs when the list is
  sorted (or near sorted)!

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Quicksort Complexity Analysis

• The average case must be betweenthe best case
  O(n log n) and the worst case is O(n2).
• Analysis yields a complex recurrence relation.
• The average case number of comparisons turns out
  to be approximately 1.386nlog n 2.846n.
• Therefore the average case time complexity isO(n
  log n).

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Quicksort Complexity Analysis

• Best case O(n log n)
• Worst case O(n2)
• Average case O(n log n)
• Note that the quick sort is inferior to insertion
  sort and merge sort if the list is sorted, nearly
  sorted, or reverse sorted.

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READING

• READ Chapter 13 14
• Chapters 15 (Dynamic Programming), 16
  (Backtracking) and 17 (Branch and Bound) are
  useful algorithm design methods and you should
  read and try to understand them
• The final exam will mostly cover (about 90) from
  the second half materials