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The Planar Point Location Problem

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Lecture’s Road Map

Lecture’s Road Map

I/O-efficient Point Location using Persistent B-TreesLars Arge, Andrew Danner, and Sha-Mayn TehDepartment of Computer Science, Duke University (2003)

Michal Balas

The Planar Point Location Problem

Storing a planar subdivision defined by N line segments such that the region containing a query point p can be computed efficiently

Michal Balas

Planar Point Location Applications

- Geographic Information systems (GIS)
- Spatial Databases
- Graphics
Usually the datasets are larger than the size of physical memory and must reside on disk

Michal Balas

Previous Works

- So far, few theoretically I/O efficient structures were developed, but all are relatively complicated and none of them was implemented
- Vahrenhold and Hinrichs (2001) suggested a heuristic structure that is simple and efficient but theoretically non optimal

Michal Balas

Goal

find a planar point location structure that minimizes the number of I/Os needed to answer a query, which is efficient both in theory and in practice.

Michal Balas

Lecture’s Road Map

- Motivation
- The Vertical Ray Shooting problem and the need of persistent data structures
- Review:
- B-trees, B+ trees, and I/O model
- Persistent B-trees

- The modified Persistent B-tree
- Experimental results
- Open problems

Michal Balas

Vertical Ray Shooting

- A generalized version of the Planar Point Location problem
- Given a set of N non-intersecting segments in the plane, construct a data structure such that the segment directly above a query point p can be found efficiently
- We will consider this problem.

Michal Balas

Example

Michal Balas

Vertical Ray Shooting

- Based on the persistent search tree idea of Sarnak and Tarjan (1986).
- Any vertical line l in the plane introduces an “above-below” order on the segments it intersects.
- We will “sweep” the plane from left to right with a vertical line
- Our “critical” x-axis points are the endpoints of all segments

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sort critical points by x-values
- For each critical point pi=(xi,yi) we can build a search tree for the segments intersecting a vertical line at xi according to the y-values (at xi)
- Until the next critical point pi+1 the tree is static – it will change only in the next begin/end point of a segment

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Worst case analysis:
- Hold a search tree to each critical point
- Space: O(n2)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- We should use the fact that two consecutive trees (versions) differ only by one insertion or deletion (assuming distinct x-values for all endpoints).

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Persistent data structure
- Preserves versions. In ordinary (ephemeral) data structures there is only one last version (every update changes the data structure so its state before the update can no longer be accessed)
- Each update creates a version
- The current version of the structure can be modified and all versions of the structure, past and present, can be accessed.

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- We would like to save a version of the search tree for each critical point. Since we want to be space efficient, we will use persistent search tree.
- A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed.
- Here the persistent search tree should supports insertions and deletions in the present and queries in the past. (partially persistent)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- We will insert a segment into the persistent search tree when its left endpoint is encountered
- We will delete a segment persistently from the tree when its right endpoint is encountered.
- Two consecutive versions of the tree differ only by a certain number of deletions and insertions (in the distinct x-values case by 1 only)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Given a query point p=(x,y) , we will search for the position of y in the version of the search tree when the sweep line was at x.

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Path Copying:
- A balanced search tree
- When x is inserted the changes are only on the path from the root to x
- Instead of copying the whole tree we will copy only the updated path
- The roots will be ordered by version

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Path Copying:
- Space: O(nlogn) – better, but not good enough

r1

r2

x

Michal Balas

left

t1

t2

Vertical Ray Shooting & Persistent Search Trees- Extra Pointers :
- Instead of copying the path, we will save for each node a few pointers ( a list of left children and right children, thought it’s a binary tree)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Extra Pointers :
- Here there is no limitation on the # of pointers per node
- In the worst case, it will take O(logn) time to find the relevant version per node (the pointers are in a binary search tree) – which is not optimal
- We need constant time per node

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution:
- Limited node copying, k extra pointers per node
- k should be a small positive number (k=1 will do)
- When a pointer is added to a node, if there is no empty slot for a new pointer, we copy the node, setting the initial left and right pointers of the copy to their latest values.
- Update the parent with the new copy, if the parent has no free slot the process is repeated.

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution - Space analysis
- Amortized analysis: we will see that every set of m operations takes O(m) space.
- The potential of the structure is defined to be:
F = # live nodes – (1/k)*(# free slots in the live nodes)

- amortized space cost of update = (actual # of nodes it creates) – DF

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution - Space analysis
- We will show that amortized space cost of an update is bounded by O(1) per update.
- If a new unused slot in node v is used, but the node is still not full, then the actual # of new nodes created is 0, DF is (-1/k) (#free slots in live nodes decreased by 1), thus amortized space cost of this update is 1/k.
- If node copying has occurred, the actual # of new nodes created is 1, DF is 1 (#free slots in live nodes increased by k), thus amortized space cost of this update is 0.

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution - Space analysis
- During an update, node copying continues in the path from node to root until the root is copied or a node with a free slot is reached.
- The amortized space cost of node copying is 0 and of occupying a free slot is 1/k

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution - Space analysis
- The total amortized space cost of an update is constant (0 or 1/k)
- The space of rebalance information per node is constant
- In red-black trees, rebalancing after deletion or insertion can be done in O(1) rotations and O(1) color changes per update in the amortized case
- Since an insertion or deletion requires O(1) new pointers not counting node copying, the amortized space cost of an update is O(1)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution - Space analysis
- sum up over all updates:
amortized space cost over all updates

= cn = required space – (Fend – Fstart)

- Fstart=0 (we start with an empty data structure)
- Fend=O(n) (according to the potential function definition, this is an upper bound on the potential in the end)
- Required space = cn + O(n) = O(n) (this is a bound on the number of nodes created)

- sum up over all updates:

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- Sarnak & Tarjan solution – Complexity
- O(log m) query time (m is the total # of updates)
- O(log n) update time (n is the current size of the set)
- O(1) amortized space per update
- O(nlogn) preprocessing time

Michal Balas

Where are we going?

The use of Persistent Data structures

(always preserves the previous version of itself when it is modified)

The use of B-trees in the I/O Model

(B-tree is the I/O model equivalent of a search tree)

I/O efficient Persistent B-tree

(works great with totally ordered elements)

Modified I/O efficient Persistent B-tree

(only elements present in the same version of the structure need to

be comparable)

Michal Balas

28

Vertical Ray Shooting & Persistent Search Trees

- Two segments that cannot be intersected with the same vertical line are not comparable ( “above-below”)
- Corollary: Not all segments stored in the persistent structure over its lifespan are comparable

An I/O efficient structure cannot directly be obtained using a persistent B-tree (because standard persistent B-trees require total order on all elements)

Michal Balas

Vertical Ray Shooting & Persistent Search Trees

- To make the structure I/O-efficient, we need to modify the tree so it will only require elements present in the same version of the structure to be comparable

Michal Balas

Lecture’s Road Map

- Motivation
- The Vertical Ray Shooting problem and the need of persistent data structures
- Review:
- B-trees, B+ trees, and I/O model
- Persistent B-trees

- The modified Persistent B-tree
- Experimental results
- Open problems

Michal Balas

Review: The I/O Model

- Infinite disk size
- M - Main Memory size
- B - Block size
- N - elements in the structure

D

M

Block I/O

Michal Balas

Review: The I/O Model - Cont

- Computation can only occur on data stored in main memory.
- We are interested in the number of I/Os used to answer a query.
- The B-tree is the external memory equivalent of the balanced search tree in internal memory.

Michal Balas

Review: B-tree

- A balanced search tree
- All leaves are on the same level
- All internal nodes (except the root) have between B/2 and B children (q(B))
- A node/leaf can be stored in O(1) blocks

Michal Balas

Review: B-tree - Cont

- Space complexity of the tree: O(N/B) blocks (where N is the number of elements) – linear
- Tree height: O(logBN)
- Insert/Delete can be done with O(logBN) I/Os

Michal Balas

Review: B+-tree

- It is a B-tree in which all elements are stored in the leaves.
- The internal nodes contain “routing elements”.

Michal Balas

Where are we going?

The use of Persistent Data structures

(always preserves the previous version of itself when it is modified)

The use of B-trees in the I/O Model

(B-tree is the I/O model equivalent of a search tree)

I/O efficient Persistent B-tree

(works great with totally ordered elements)

Modified I/O efficient Persistent B-tree

(only elements present in the same version of the structure need to

be comparable)

Michal Balas

38

Review: Persistent B-tree

- Directed acyclic graph
- The elements are in the sinks (leaves)
- “routing elements” in internal nodes

- Elements (and nodes) augmented with “existence interval”
- In this interval the element is “alive”
- An element is “alive” - between its insert and its delete version

Michal Balas

Review: Persistent B-tree - Cont

- Nodes “alive” at time t form a (aB,B) B-tree, 0<a<1/2
- We will work with a=1/4

- Additional invariant:
- A new node must contain between (a+g)B and (1-g)B alive elements ( a > g )
- For g=1/8, a=1/4,new node contains between (3/8)B and (7/8)B alive elements
- We require that g>2/B, a-g >=1/B, 2a+3g<= 1-3/B

Michal Balas

Review: Persistent B-tree - Cont

- In order to find the appropriate root at time t, the roots are stored in a standard B-tree
- Takes O(logBN) I/Os

- A node/leaf contains O(B) elements = O(1) blocks

# Blocks needed to hold the structure: O(N/B)

Michal Balas

Persistent B-tree Insert

- x is the element to insert into the current version of the tree
- Search the leaf l and insert x (O(logBN) I/Os)
- if l contains > B elements -> Block overflow
- Version-Split (copy all k alive elements from l to a new node v and mark l as dead)
- If k is in [(3/8)B,(7/8)B] - simple
- If k > (7/8)B – strong overflow
- If k < (3/8)B – strong underflow
Strong overflow/underflow violates the additional invariant we defined earlier

Michal Balas

Persistent B-tree Insert

- If k is in [(3/8)B,(7/8)B] :
recursively update parent(l): persistently delete the reference to l and insert a reference to v

Michal Balas

Persistent B-tree Insert - Cont

- If k > (7/8)B – strong overflow
- split
create nodes v1, v2 each with k/2 elements.

k/2 is in ((3/8)B,(7/8)B) (this is not tight)

- Update parent(l) recursively: persistently delete the reference to l and insert two references to v1, v2

Michal Balas

Persistent B-tree Insert - Cont

- If k < (3/8)B – strong underflow
- Version-split of sibling l’ of l -> obtain k’ other alive elements (k’ is in [aB,B])
k+k’>= 2aB, and a > g, thus k+k’ > (a+g)B (the invariant…)

1) ifk+k’ <= (1-g)B: merge -create a new leaf with k+k’ elements

2) if k+k’ >(1-g)B: share – split to create two new leaves.

- Update parent(l) recursively: persistently delete two references and insert one or two

Michal Balas

Persistent B-tree Delete

- x is the element to delete from the current version of the tree
- Search the leaf l that contains and mark x as dead (O(logBN) I/Os)
- if l contains < (1/4)B alive elements -> Block underflow (this is also a strong underflow, since k < (3/8)B )
- Version-Split on a sibling node to obtain k+k’ elements.
k+k’>= 2aB -1 , and a- g > =1/B, thus k+k’ > (a+g)B (the invariant…)

mark l dead and create a new node v with k+k’ elements (merge)

if there is a strong overflow in v – share (as in insert)

- Version-Split on a sibling node to obtain k+k’ elements.
- Update parent(l) recursively: persistently delete two references and insert one or two references

Michal Balas

Persistent B-tree – Rebalance Operations

Delete

Insert

Block Overflow

Block Underflow

Done

0,0

Version-split

Version-split

Done

-1,+1

Strong Underflow

Strong Overflow

Merge

Split

Done

-2,+1

Done

-1,+2

Strong Overflow

Split

Done

-2,+2

Michal Balas

Persistent B-tree - Complexity

- Updates: O(logBN) I/Os
- search and rebalance on one path from root to leaf

- What about the required space?

Michal Balas

Persistent B-tree - Complexity

- A few observations:
- A rebalance operation on leaf creates <= 2 new nodes
- Once a leaf is created, at least gB updates have to be performed on it before another rebalance operation will occur.
- Two version-splits might only create one new leaf
- Each time a leaf is created or a leaf version-split performed, a corresponding insertion or deletion is performed recursively one level up the tree.

- During N updates:
- # leaves created <= 2N/gB = O(N/B)
- # leaf version-splits<= 2N/gB
- # nodes created one level up the tree <= 22N/(gB)2
- By induction: # nodes created i levels up the tree <= 2i+1N/(gB)i+1
- Total # nodes created <=
(it is also the # of blocks used after N updates)

- Space: O(N/B) blocks

Michal Balas

Lecture’s Road Map

- Motivation
- The Vertical Ray Shooting problem and the need of persistent data structures
- Review:
- B-trees, B+ trees, and I/O model
- Persistent B-trees

- The modified Persistent B-tree
- Experimental results
- Open problems

Michal Balas

Where are we going?

The use of Persistent Data structures

(always preserves the previous version of itself when it is modified)

The use of B-trees in the I/O Model

(B-tree is the I/O model equivalent of a search tree)

I/O efficient Persistent B-tree

(works great with totally ordered elements)

Modified I/O efficient Persistent B-tree

(only elements present in the same version of the structure need to

be comparable)

Michal Balas

51

The modified Persistent B-tree

- Why do we need to modify the standard Persistent B-tree?
- Before, a few facts about standard B-tree:
- The elements are in the leaves
- Internal nodes contain “routing elements”
- When a node v is created a reference is added to parent(v) – normally a copy of the maximal element in v is used as a routing element in parent(v)

Michal Balas

The modified Persistent B-tree

The structure contains multiple live copies of the same element.

There may be copies of an element as routing elements long after the element is deleted

When searching for an element in the structure at version t we might be comparing to a copy of a dead element.

Michal Balas

The modified Persistent B-tree

In this application (vertical ray shooting) not all elements (segments) stored in the data structure during its entire lifespan are above-below comparable

We cannot use the standard version of a persistent B-tree, since it requires all elements in the structure to be comparable.

Modification is needed!

Michal Balas

The modified Persistent B-tree

- We want the structure to only require elements present in the same version to be comparable
- The modified structure:
- Alive elements in time t form a B-tree with elements in all nodes - internal + leaves. (not just in leaves)
- # live copies of an element at any given time t <= 1

Michal Balas

The modified Persistent B-tree

- There will be some modification to the rebalance operations
- The Insert algorithm remains
- The delete algorithm is slightly modified:

Michal Balas

The modified Persistent B-tree

- The modified delete algorithm:
When deleting an element x which is in internal node u we need to be careful since x is associated with a reference to a child ucof u that is still alive

- Find y : the predecessor of x in a leaf below u
- Persistently delete y
- Persistently delete x from u
- Insert a live copy of y with a reference to the child uc
- Perform the needed rebalance operations

Michal Balas

The modified Persistent B-tree- rebalance operations

- Version-Split: copying all alive elements of u to a new node v

x

x

u

u

v

We can use x as the element associated with the reference to the new node v , since the elements in v are a subsets of the elements in u

Michal Balas

The modified Persistent B-tree- rebalance operations

- Split: when a strong overflow occurs after a version-split of u, two new nodes v, v’ are created

x

y

x

u

y

u

v

v’

we promote the maximal element y in v to be associated with the reference to v in parent(u) (instead of storing y in v).

x will be associated with the reference to v’ in parent(u).

v has one less element than it would have had using the regular split, but O(gB) updates are still required on v before further structural changes are needed

Michal Balas

The modified Persistent B-tree- rebalance operations

- Merge: when a strong underflow occurs after a version-split of u, a version-split of u’s sibling u’ is performed, and a new node v is created with the alive elements from u,u’

x

y

y

u

u’

v

u

u’

X

The maximal between x and y , say y, is used as the reference to the new node v. x is demoted and stored in the new node v

v has one more element than it would have had using the regular merge. But as in split, O(gB) updates are still needed on v before further structural changes are needed

Michal Balas

The modified Persistent B-tree- rebalance operations

- Share: when a merge would result in a new node with a strong overflow, instead a version-split on the two sibling nodes u and u’ is performed, and two new nodes v, v’ are created.

x

y

z

y

u

u’

z

v

u

u’

v’

X

The maximal element y can be reused as the reference to v’ but x cannot be used as a reference to v.x is demoted to v and the maximal element z in v is promoted to parent(u).

# of elements in the new node v is identical to the # of elements we would have had using the regular share.

Michal Balas

The modified Persistent B-tree- Complexity

- Even though there is a difference in the number of elements, the previous space arguments still apply
- Space: O(N/B) blocks
- Update on the newest version: O(logBN) I/Os

Michal Balas

The modified Persistent B-tree- Summary

- A set of N non-intersecting segments in the plane can be processed into a data structure of size O(N/B) in O(NlogBN) I/Os such that a vertical ray-shooting query can be answered in O(logBN) I/Os

Michal Balas

The modified Persistent B-tree- Summary

N updates on a persistent B-tree (standard or modified) takes I/Os

Goodrich et al. showed how to construct a persistent B-tree structure (different from the basic one described earlier) in I/Os (the sorting bound)

The structure by Goodrich et al., requires that all elements in the structure over its lifespan are comparable

In the modified tree we cannot use that, since the elements are not totally ordered, so this construction complexity is not reached (so far)

Michal Balas

64

Where have we come from?

The use of Persistent Data structures

(always preserves the previous version of itself when it is modified)

The use of B-trees in the I/O Model

(B-tree is the I/O model equivalent of a search tree)

I/O efficient Persistent B-tree

(works great with totally ordered elements)

Modified I/O efficient Persistent B-tree

(only elements present in the same version of the structure need to

be comparable)

Michal Balas

65

- Motivation
- The Vertical Ray Shooting problem and the need of persistent data structures
- Review:
- B-trees, B+ trees, and I/O model
- Persistent B-trees

- The modified Persistent B-tree
- Experimental results
- Open problems

Michal Balas

Experimental Results

- Compared the persistent B-tree and the grid structure of Vahrenhold and Hinrichs
- Implemented both using TPIE library
- Used road data , containing all roads in the US. Roads are broken at intersections
- The query points were randomly sampled from the datasets
- Used also worst case artificially generated dataset

Michal Balas

Experimental Results

- In terms of query efficiency:
- # I/Os per query, Time per query – both are much lower in the persistent B-tree than in the Grid structure. In synthetically generated worst case dataset B-tree uses significantly fewer I/Os

- size, Construction efficiency – grid construction algorithm outperforms the persistent B-tree on the real life datasets, though on the worst case dataset the persistent B-tree was significantly better.

Michal Balas

- Motivation
- The Vertical Ray Shooting problem and the need of persistent data structures
- Review:
- B-trees, B+ trees, and I/O model
- Persistent B-trees

- The modified Persistent B-tree
- Experimental results
- Open problems

Michal Balas

Open Problems

- One major open problem is to construct the structure in I/Os (here we saw a trivial algorithm that constructs in

Michal Balas

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