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Algorithms and Data Structures Lecture III. Simonas Šaltenis Aalborg University [email protected] This Lecture. Divide-and-conquer technique for algorithm design. Example problems: Tiling Searching ( binary search ) Sorting ( merge sort ) . A tiling of the board with trominos:. Tiling.

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this lecture
This Lecture
  • Divide-and-conquer technique for algorithm design. Example problems:
    • Tiling
    • Searching (binary search)
    • Sorting (merge sort).
tiling

A tiling of the board with trominos:

Tiling

A tromino tile:

And a 2nx2n board with a hole:

tiling trivial case n 1
Tiling: Trivial Case (n = 1)
  • Trivial case (n = 1): tiling a 2x2 board with a hole:
  • Idea – try somehow to reduce the size of the original problem, so that we eventually get to the 2x2 boards which we know how to solve…
tiling dividing the problem
Tiling: Dividing the Problem
  • To get smaller square boards let’s divide the original board into for boards
  • Great! We have one problem of the size 2n-1x2n-1!
  • But: The other three problems are not similar to the original problems – they do not have holes!
tiling dividing the problem1
Tiling: Dividing the Problem
  • Idea: insert one tromino at the center to get three holes in each of the three smaller boards
  • Now we have four boards with holes of the size 2n-1x2n-1.
  • Keep doing this division, until we get the 2x2 boards with holes – we know how to tile those
tiling algorithm
Tiling: Algorithm

INPUT: n – the board size (2nx2n board), L – location of the hole.

OUTPUT: tiling of the board

Tile(n, L)

if n = 1 then

Trivial case

Tile with one tromino

return

Divide the board into four equal-sized boards

Place one tromino at the center to cut out 3 additional holes

Let L1, L2, L3, L4 denote the positions of the 4 holes

Tile(n-1, L1)

Tile(n-1, L2)

Tile(n-1, L3)

Tile(n-1, L4)

divide and conquer
Divide and Conquer
  • Divide-and-conquer method for algorithm design:
    • If the problem size is small enough to solve it in a straightforward manner, solve it. Else:
      • Divide: Divide the problem into two or more disjoint subproblems
      • Conquer: Use divide-and-conquer recursively to solve the subproblems
      • Combine: Take the solutions to the subproblems and combine these solutions into a solution for the original problem
tiling divide and conquer
Tiling: Divide-and-Conquer
  • Tiling is a divide-and-conquer algorithm:
    • Just do it trivially if the board is 2x2, else:
    • Divide the board into four smaller boards (introduce holes at the corners of the three smaller boards to make them look like original problems)
    • Conquer using the same algorithm recursively
    • Combine by placing a single tromino in the center to cover the three introduced holes
binary search
Binary Search
  • Find a number in a sorted array:
    • Just do it trivially if the array is of one element
    • Else divide into two equal halves and solve each half
    • Combine the results

INPUT: A[1..n] – a sorted (non-decreasing) array of integers, s – an integer.

OUTPUT: an index j such that A[j] = s. NIL, if "j (1£j£n): A[j] ¹ s

Binary-search(A, p, r, s):

if p = r then

if A[p] = s then return p

else return NIL

q¬ë(p+r)/2û

ret ¬ Binary-search(A, p, q, s)

if ret = NIL then

return Binary-search(A, q+1, r, s)

else return ret

recurrences
Recurrences
  • Running times of algorithms with Recursive calls can be described using recurrences
  • A recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs
  • Example: Binary Search
binary search improved
Binary Search (improved)

INPUT: A[1..n] – a sorted (non-decreasing) array of integers, s – an integer.

OUTPUT: an index j such that A[j] = s. NIL, if "j (1£j£n): A[j] ¹ s

Binary-search(A, p, r, s):

if p = r then

if A[p] = s then return p

else return NIL

q¬ë(p+r)/2û

if A[q] £ s then return Binary-search(A, p, q, s)

else return Binary-search(A, q+1, r, s)

  • T(n) = Q(n) – not better than brute force!
  • Clever way to conquer:
    • Solve only one half!
running time of bs
Running Time of BS
  • T(n) = Q(lg n) !
merge sort algorithm
Merge Sort Algorithm
  • Divide: If S has at least two elements (nothing needs to be done if S has zero or one elements), remove all the elements from S and put them into two sequences, S1 and S2 , each containing about half of the elements of S. (i.e. S1 contains the firstén/2ù elements and S2 contains the remaining ën/2û elements).
  • Conquer: Sort sequences S1 and S2 using Merge Sort.
  • Combine: Put back the elements into S by merging the sorted sequences S1 and S2 into one sorted sequence
merge sort algorithm1
Merge Sort: Algorithm

Merge-Sort(A, p, r)

if p < r then

q¬ ë(p+r)/2û

Merge-Sort(A, p, q)

Merge-Sort(A, q+1, r)

Merge(A, p, q, r)

Merge(A, p, q, r)

Take the smallest of the two topmost elements of sequences A[p..q] and A[q+1..r] and put into the resulting sequence. Repeat this, until both sequences are empty. Copy the resulting sequence into A[p..r].

merge sort summarized
Merge Sort Summarized
  • To sort n numbers
    • if n=1 done!
    • recursively sort 2 lists of numbers ën/2û and én/2ù elements
    • merge 2 sorted lists in Q(n) time
  • Strategy
    • break problem into similar (smaller) subproblems
    • recursively solve subproblems
    • combine solutions to answer
running time of mergesort
Running time of MergeSort
  • Again the running time can be expressed as a recurrence:
repeated substitution method
Repeated Substitution Method
  • Let’s find the running time of merge sort (let’s assume that n=2b, for some b).
example finding min and max
Example: Finding Min and Max
  • Given an unsorted array, find a minimum and a maximum element in the array

INPUT: A[l..r] – an unsorted array of integers, l £ r.

OUTPUT: (min, max) such that "j (l£j£r): A[j] ³min and A[j] £max

MinMax(A, l, r):

if l = r then return (A[l], A[r]) Trivial case

q¬ ë(l+r)/2û Divide

(minl, maxl)¬ MinMax(A, l, q)

(minr, maxr)¬ MinMax(A, q+1, r)

if minl < minr then min = minl else min = minr

if maxl > maxr then max = maxl else max = maxr

return (min, max)

Conquer

Combine

next week
Next Week
  • Analyzing the running time of recursive algorithms (such as divide-and-conquer)
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