Study Notes_Algorithms_online course wee

the union find problem

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two classic algorithms: Quick find and Quick union

Steps to develop a usable algorithm

• model the problem
• find an algorithm to solve it
• fast enough? fits in memory?
• if not, find why.
• figure out a way to address the problem
• iterate until satisfied

The scientific method

Mathematical analysis

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• dynamic connectivity (the model of the problem for union find)

the problem: find if there is a path.(we will do "find the path" later, not today)

• modeling the connections
  • reflexive: p-p
    
  • symmetric: if p-q, then q-p
  • transitive: if p-q, q-r, then p-r
• connected components: the maximal set of objects that are mutually connected
• find query: check if two objects are in the same connected components
• union command: replace components containing two objects with their union

• quick find (eager approach)
  • Data structure: 
    • Integer array id[] of size N.
    • Interpretation: p and q are connected if they have the same id.
  • Find: Check if p and q have the same id.
  • Union: To merge components containing p and q, change all entries with id[p] to id[q].
    
  • quick find is too slow.
  • quadratic time is too slow, can't accept quadratic algorithms for large problems. the reason is they don't scale.

• quick union (lazy approach)
  • Data structure:
    • Integer array id[] of size N.
    • Interpretation: id[i] is parent of i. 
    • Root of i is id[id[id[...id[i]...]]].
  • Find. Check if p and q have the same root.
  • Union. Set the id of q's root to the id of p's root.
  • Quick union is also too slow
    • Quick-find defect.
      • Union too expensive (N steps).
      • Trees are flat, but too expensive to keep them flat.
    • Quick-union defect.
      • Trees can get tall.
      • Find too expensive (could be N steps)
      • Need to do find to do union
  • What is the maximum number of array accesses during a find operation when using the quick-union data structure on N elements?---Linear
    

• improvements
  • weighting
    • Weighted quick-union.
      • Modify quick-union to avoid tall trees.
      • Keep track of size of each component.
      • Balance by linking small tree below large one.
    • Java implementation.
      • Almost identical to quick-union.
      • Maintain extra array sz[] to count number of elements in the tree rooted at i.
    • Find. Identical to quick-union.
    • Union. Modify quick-union to merge smaller tree into larger tree update the sz[] array.
    • Analysis.
      • Find: takes time proportional to depth of p and q.
      • Union: takes constant time, given roots.
      • Fact: depth is at most lg N. [needs proof]
  • Path compression
    
    • Path compression. Just after computing the root of i,set the id of each examined node to root(i).
    • Standard implementation: add second loop to root() to set the id of each examined node to the root.
    • Simpler one-pass variant: make every other node in path point to its grandparent.
  • WQUPC
    • Theorem. Starting from an empty data structure, any sequence of M union and find operations on N objects takes O(N + M lg* N) time.
    • Proof is very difficult.
    • But the algorithm is still simple!
      
    • Linear algorithm?
      • Cost within constant factor of reading in the data.
      • In theory, WQUPC is not quite linear. 
      • In practice, WQUPC is linear.
    • Amazing fact: In theory, no linear linking strategy exists

• Union-find applications
  • Network connectivity.
  • Percolation. A model for many physical systems
    • N-by-N grid.
    • Each square is vacant or occupied.
    • Grid percolates if top and bottom are connected by vacant squares.
  • Image processing.
  • Least common ancestor.
  • Equivalence of finite state automata.
  • Hinley-Milner polymorphic type inference.
  • Kruskal's minimum spanning tree algorithm. 
  • Games (Go, Hex)
  • Compiling equivalence statements in Fortran.