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CSE 326: Data Structures Part 10 Advanced Data Structures

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Outline

- Multidimensional search trees
- Range Queries
- k-D Trees
- Quad Trees
- Randomized Data Structures & Algorithms
- Treaps
- Primality testing
- Local search for NP-complete problems

2,5

8,4

4,2

3,6

9,1

4,4

1,9

8,2

5,7

Multi-D Search ADT- Dictionary operations
- create
- destroy
- find
- insert
- delete
- range queries
- Each item has k keys for a k-dimensional search tree
- Searches can be performed on one, some, or all the keys or on ranges of the keys

Applications of Multi-D Search

- Astronomy (simulation of galaxies) - 3 dimensions
- Protein folding in molecular biology - 3 dimensions
- Lossy data compression - 4 to 64 dimensions
- Image processing - 2 dimensions
- Graphics - 2 or 3 dimensions
- Animation - 3 to 4 dimensions
- Geographical databases - 2 or 3 dimensions
- Web searching - 200 or more dimensions

Range Query

A range query is a search in a dictionary in which the exact key may not be entirely specified.

Range queries are the primary interface

with multi-D data structures.

Search for items based on just one key

Search for items based on ranges for all keys

Search for items based on a function of several keys: e.g., a circular range query

Range Query Examples:Two DimensionsFind in a k-D Tree

find(<x1,x2, …, xk>, root) finds the node which has the given set of keys in it or returns null if there is no such node

Node find(keyVector keys,

Node root) {

int dim = root.dimension;

if (root == NULL)

return NULL;

else if (root.keys == keys)

return root;

else if (keys[dim] < root.keys[dim])

return find(keys, root.left);

else

return find(keys, root.right);

}

runtime:

x

2-D Range Querying in 2-D TreesSearch every partition that intersects the rectangle.

Check whether each node (including leaves) falls into the range.

Range Query in a 2-D Tree

print_range(int xlow, xhigh, ylow, yhigh, Node root) {

if (root == NULL) return;

if ( xlow <= root.x && root.x <= xhigh &&

ylow <= root.y && root.y <= yhigh ){

print(root);

if ((root.dim == “x” && xlow <= root.x ) ||

(root.dim == “y” && ylow <= root.y )) print_range(root.left);

if ((root.dim == “x” && root.x <= xhigh) ||

(root.dim == “y” && root.y <= yhigh) print_range(root.right);

}

runtime: O(N)

Range Query in a k-D Tree

print_range(int low[MAXD], high[MAXD], Node root) {

if (root == NULL) return;

inrange = true;

for (i=0; i<MAXD;i++){

if ( root.coord[i] < low[i] ) inrange = false;

if ( high[i] < root.coord[i] ) inrange = false; }

if (inrange) print(root);

if ((low[root.dim] <= root.coord[root.dim] ) print_range(root.left);

if (root.coord[root.dim] <= high[root.dim])

print_range(root.right);

}

runtime: O(N)

Other Shapes for Range Querying

y

x

Search every partition that intersects the shape (circle).

Check whether each node (including leaves) falls into the shape.

insert(<5,0>)

insert(<6,9>)

insert(<9,3>)

insert(<6,5>)

insert(<7,7>)

insert(<8,6>)

k-D Trees Can Be Inefficient(but not when built in batch!)5,0

6,9

9,3

6,5

7,7

8,6

suck factor:

insert(<5,0>)

insert(<6,9>)

insert(<9,3>)

insert(<6,5>)

insert(<7,7>)

insert(<8,6>)

k-D Trees Can Be Inefficient(but not when built in batch!)5,0

6,9

9,3

6,5

7,7

8,6

suck factor: O(n)

keys

value

y

Quad Trees- Split on all (two) dimensions at each level
- Split key space into equal size partitions (quadrants)
- Add a new node by adding to a leaf, and, if the leaf is already occupied, split until only one node per leaf

quad tree node

quadrant

0,1

1,1

Center:

0,0

1,0

0,0

1,0

0,1

1,1

Quadrants:

Center

Find in a Quad Tree

find(<x, y>, root) finds the node which has the given pair of keys in it or returns quadrant where the point should be if there is no such node

Node find(Key x, Key y, Node root) {

if (root == NULL)

return NULL; // Empty tree

if (root.isLeaf())

return root; // Key may not actually be here

int quad = getQuadrant(x, y, root);

return find(x, y, root.quadrants[quad]);

}

Compares against center; always makes the same choice on ties.

runtime: O(depth)

2-D Range Query in a Quad Tree

print_range(int xlow, xhigh, ylow, yhigh, Node root){

if (root == NULL) return;

if ( xlow <= root.x && root.x <= xhigh &&

ylow <= root.y && root.y <= yhigh ){

print(root);

if (xlow <= root.x && ylow <= root.y)

print_range(root.lower_left);

if (xlow <= root.x && root.y <= yhigh)

print_range(root.upper_left);

if (root.x <= x.high && ylow <= root.x)

print_range(root.lower_right);

if (root.x <= xhigh && root.y <= yhigh)

print_range(root.upper_right);

}

runtime: O(N)

Find in a Quad Tree

find(<x, y>, root) finds the node which has the given pair of keys in it or returns quadrant where the point should be if there is no such node

Node find(Key x, Key y, Node root) {

if (root == NULL)

return NULL; // Empty tree

if (root.isLeaf())

return root; // Key may not actually be here

int quad = getQuadrant(x, y, root);

return find(x, y, root.quadrants[quad]);

}

Compares against center; always makes the same choice on ties.

runtime: O(depth)

Delete Example

delete(<10,2>)(i.e., c)

a

b

c

a

g

d

d

e

f

e

g

f

- Find and delete the node.
- If its parent has just one child, delete it.
- Propagate!

b

c

Nearest Neighbor Search

getNearestNeighbor(<1,4>)

a

b

c

a

g

d

e

d

e

f

g

f

- Find a nearby node (do a find).
- Do a circular range query.
- As you get results, tighten the circle.
- Continue until no closer node in query.

b

c

Works on

k-D Trees, too!

Quad Trees vs. k-D Trees

- k-D Trees
- Density balanced trees
- Number of nodes is O(n) where n is the number of points
- Height of the tree is O(log n) with batch insertion
- Supports insert, find, nearest neighbor, range queries
- Quad Trees
- Number of nodes is O(n(1+ log(/n))) where n is the number of points and is the ratio of the width (or height) of the key space and the smallest distance between two points
- Height of the tree is O(log n + log )
- Supports insert, delete, find, nearest neighbor, range queries

To Do

- Read (a little) about k-D trees in Weiss 12.6

Pick a Card

Warning! The Queen of Spades is a very unlucky card!

Randomized Data Structures

- We’ve seen many data structures with good average case performance on random inputs, but bad behavior on particular inputs
- Binary Search Trees
- Instead of randomizing the input (since we cannot!), consider randomizing the data structure
- No bad inputs, just unlucky random numbers
- Expected case good behavior on any input

What’s the Difference?

- Deterministic with good average time
- If your application happens to always use the “bad” case, you are in big trouble!
- Randomized with good expected time
- Once in a while you will have an expensive operation, but no inputs can make this happen all the time
- Kind of like an insurance policy for your algorithm!

Treap Dictionary Data Structure

heap in yellow; search tree in blue

- Treaps have the binary search tree
- binary tree property
- search tree property
- Treaps also have the heap-order property!
- randomly assigned priorities

2

9

6

7

4

18

7

8

9

15

10

30

Legend:

priority

key

15

12

Treap Insert

- Choose a random priority
- Insert as in normal BST
- Rotate up until heap order is restored (maintaining BST property while rotating)

insert(15)

2

9

2

9

2

9

6

7

14

12

6

7

14

12

6

7

9

15

7

8

7

8

9

15

7

8

14

12

insert(8)

insert(9)

insert(12)

6

7

6

7

2

9

2

9

7

8

6

7

6

7

15

12

7

8

7

8

Tree + Heap… Why Bother?Insert data in sorted order into a treap; what shape tree comes out?

Legend:

priority

key

Treap Delete

delete(9)

2

9

6

7

rotate left

rotate left

- Find the key
- Increase its value to
- Rotate it to the fringe
- Snip it off

6

7

9

15

9

7

8

15

12

7

8

9

15

6

7

rotate right

15

12

7

8

9

9

15

15

12

Treap Summary

- Implements Dictionary ADT
- insert in expected O(log n) time
- delete in expected O(log n) time
- find in expected O(log n) time
- but worst case O(n)
- Memory use
- O(1) per node
- about the cost of AVL trees
- Very simple to implement, little overhead – less than AVL trees

Other Randomized Data Structures & Algorithms

- Randomized skip list
- cross between a linked list and a binary search tree
- O(log n) expected time for finds, and then can simply follow links to do range queries
- Randomized QuickSort
- just choose pivot position randomly
- expected O(n log n) time for any input

Randomized Primality Testing

- No known polynomial time algorithm for primality testing
- but does not appear to be NP-complete either – in between?
- Best known algorithm:
- Guess a random number 0 < A < N
- If (AN-1 % N) 1, then N is not prime
- Otherwise, 75% chance N is prime
- or is a “Carmichael number” – a slightly more complex test rules out this case
- Repeat to increase confidence in the answer

Randomized Search Algorithms

- Finding a goal node in very, very large graphs using DFS, BFS, and even A* (using known heuristic functions) is often too slow
- Alternative: random walk through the graph

N-Queens Problem

- Place N queens on an N by N chessboard so that no two queens can attack each other
- Graph search formulation:
- Each way of placing from 0 to N queens on the chessboard is a vertex
- Edge between vertices that differ by adding or removing one queen
- Start vertex: empty board
- Goal vertex: any one with N non-attacking queens (there are many such goals)
- Demo

Random Walk – Complexity?

- Random walk – also known as an “absorbing Markov chain”, “simulated annealing”, the “Metropolis algorithm” (Metropolis 1958)
- Can often prove that if you run long enough will reach a goal state – but may take exponential time
- In some cases can prove that with high probability a goal is reached in polynomial time
- e.g., 2-SAT, Papadimitriou 1997
- Widely used for real-world problems where actual complexity is unknown – scheduling, optimization

Traveling Salesman

Recall the Traveling Salesperson (TSP) Problem: Given a fully connected, weightedgraph G = (V,E), is there a cycle that visits all vertices exactly once and has total cost K?

- NP-complete: reduction from Hamiltonian circuit
- Occurs in many real-world transportation and design problems
- Randomized simulated annealing algorithm demo

Latin Squares

- Randomization can be combined with depth first search
- When a branch of the search terminates without finding a solution, algorithm backs up to the last choice point: backtracking search
- Instead of make choice of branch to follow systematically, make it randomly
- If your random choices are unlucky, give up and start over again
- Demo

Be Sure to Bring

- 1 page of notes
- A hand calculator
- Several #2 pencils

Final Review: What you need to know

- Basic Math
- Logs, exponents, summation of series
- Proof by induction
- Asymptotic Analysis
- Big-oh, Theta and Omega
- Know the definitions and how to show f(N) is big-O/Theta/Omega of (g(N))
- How to estimate Running Time of code fragments
- E.g. nested “for” loops
- Recurrence Relations
- Deriving recurrence relation for run time of a recursive function
- Solving recurrence relations by expansion to get run time

Final Review: What you need to know

- Lists, Stacks, Queues
- Brush up on ADT operations – Insert/Delete, Push/Pop etc.
- Array versus pointer implementations of each data structure
- Amortized complexity of stretchy arrays
- Trees
- Definitions/Terminology: root, parent, child, height, depth etc.
- Relationship between depth and size of tree
- Depth can be between O(log N) and O(N) for N nodes

Final Review: What you need to know

- Binary Search Trees
- How to do Find, Insert, Delete
- Bad worst case performance – could take up to O(N) time
- AVL trees
- Balance factor is +1, 0, -1
- Know single and double rotations to keep tree balanced
- All operations are O(log N) worst case time
- Splay trees – good amortized performance
- A single operation may take O(N) time but in a sequence of operations, average time per operation is O(log N)
- Every Find, Insert, Delete causes accessed node to be moved to the root
- Know how to zig-zig, zig-zag, etc. to “bubble” node to top

Final Review: What you need to know

- Priority Queues
- Binary Heaps: Insert/DeleteMin, Percolate up/down
- Array implementation
- BuildHeap takes only O(N) time (used in heapsort)
- Binomial Queues: Forest of binomial trees with heap order
- Merge is fast – O(log N) time
- Insert and DeleteMin based on Merge
- Hashing
- Hash functions based on the mod function
- Collision resolution strategies
- Chaining, Linear and Quadratic probing, Double Hashing
- Load factor of a hash table

Final Review: What you need to know

- Sorting Algorithms: Know run times and how they work
- Elementary sorting algorithms and their run time
- Selection sort
- Heapsort – based on binary heaps (max-heaps)
- BuildHeap and repeated DeleteMax’s
- Mergesort – recursive divide-and-conquer, uses extra array
- Quicksort – recursive divide-and-conquer, Partition in-place
- fastest in practice, but O(N2) worst case time
- Pivot selection – median-of-three works best
- Know which of these are stable and in-place
- Lower bound on sorting, bucket sort, and radix sort

Final Review: What you need to know

- Disjoint Sets and Union-Find
- Up-trees and their array-based implementation
- Know how Union-by-size and Path compression work
- No need to know run time analysis – just know the result:
- Sequence of M operations with Union-by-size and P.C. is (M (M,N)) – just a little more than (1) amortized time per op
- Graph Algorithms
- Adjacency matrix versus adjacency list representation of graphs
- Know how to Topological sort in O(|V| + |E|) time using a queue
- Breadth First Search (BFS) for unweighted shortest path

Final Review: What you need to know

- Graph Algorithms (cont.)
- Dijkstra’s shortest path algorithm
- Depth First Search (DFS) and Iterated DFS
- Use of memory compared to BFS
- A* - relation of g(n) and h(n)
- Minimum Spanning trees – Kruskal’s & Prim’s algorithms
- Connected components using DFS or union/find
- NP-completeness
- Euler versus Hamiltonian circuits
- Definition of P, NP, NP-complete
- How one problem can be “reduced” to another (e.g. input to HC can be transformed into input for TSP)

Final Review: What you need to know

- Multidimensional Search Trees
- k-d Trees – find and range queries
- Depth logarithmic in number of nodes
- Quad trees – find and range queries
- Depth logarithmic in inverse of minimal distance between nodes
- But higher branching fractor means shorter depth if points are well spread out (log base 4 instead of log base 2)
- Randomized Algorithms
- expected time vs. average time vs. amortized time
- Treaps, randomized Quicksort, primality testing

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