Important Problems

• Section 1.3
• Sorting
  • This is an amusing section.
  • The sorting problem is: given a set of items, with a property by which these items can be ordered, place the items in ascending (or descending) order.
  • There must be a relation between the items which permits ordering.
  • Integers, records ...
  • Ordering is performed on a sub field called a key
  • Or possibly on multiple keys.
    • In this case, we would use a primary key for comparison
    • If the primary keys of two fields were equivalent, we would move to secondary keys
    • and so on
  • There are a number of sorting algorithms. This is by no means an exhaustive list.
  • And we continue to create new sorting algorithms all of the time.
  • Two classes
    • Comparison based - two items are compared to determine their relative position.
      • Consider one you know well, Selection Sort

      • SELECTION_SORT(A)
        
         for position ← 0 to A.size()-2  do
              min ← position
              for candidate ← position+1 to A.size()-1 do
                   if A[min] > A[candidate] then
                              min ← candidate;
              if min ≠ position then
                   swap(A[position], A[min])
         return
        
        		   

      • Note, this algorithm assumes C style arrays,
        • Positions: 0 to size -1
        • Be careful, other algorithms may not assume this.
      • This is a comparison based sort because of line 4.
      • The questions?
        • Does this algorithm work?
        • How does this algorithm work?
        • Can we argue that it is correct?
        • How efficient is it?
    • Non-Comparison based
      • Consider this algorithm

      • PERMUTATION_SORT(A)
        
         while A is not sorted do
              FindNextPermutation(A)
         return
        
        		     

      • We will study FindNextPermutation later
      • but it is possible.
      • How would you perform line 1?
      • Questions
        • Does the algorithm work?
        • How does the algorithm work?
        • Can we prove that it will work?
        • Is it efficient?
      • Why is this not a comparison sort?
  • A sort is stable if it preserves the relative order of two equivalent items.
  • A sort is in place if it does not require extra memory.
  • He states that sorting is sort of the mouse trap problem of CS
    • Everyone wants to build a better mouse trap!
• Searching
  • Given a list of items (either ordered or unordered) and a search key
  • Determine if the search key is contained in the list.
  • You know two,
    • linear or sequential search
    • and the "spectacularly efficient but limited" binary search.
• String Processing
  • Given a text (string) and a pattern (substring), locate the occurrences of the pattern in the text.
  • Find in a word processor
  • But also locate a gene in a genome
  • These tend to be really complex algorithms.
• Graph Problems
  • A graph G={V,E} is a way to represent relationships (the edges, E) between data points (the vertexes, V)
  • Roads and intersections can be represented as a graph
  • So can relationships in social media such as facebook.
  • Maps for everything but Stratego are represented by graphs.
  • Or even how functions of a program are called.
  • We will discuss the formalities of graphs when it becomes important.
  • But the problems associated with graphs are remarkable.
  • Some are quite easy, find the shortest path between two vertexes
  • Some are quite hard, Given a fully connected weighted graph, find the least cost path to visit all vertexes and return home.
• Combinatorial Problems
  • Find the optimal subset.
  • The Knapsack problem is an example of this.
  • So is the Traveling Salesperson Problem
  • As is circuit decidability
  • And many other really interesting problems.
  • We don't know how to solve many of these efficiently
  • But we can't prove that we can't solve them efficiently either.
  • This is an end of the semester topic.
• Geometric Problems
  • Directly related to geometry.
  • Given a line and a point, is the point on a line?
  • Given two line segments, do they intersect?
  • Given a set of points in the plane, what is the smallest convex polygon which encloses all those points. (Convex Hull problem)
  • Given a set of points in the plane, which are the closest (closest pair problem)
• Numerical Problems
  • Find the square root of a number.
  • Given an function, find the roots of that function.
  • Most of these problems can only be solved approximately.
  • But we have spent an enormous amount of time on these problems.
  • And it is still an active area of research, in spite of what our author implies.