Presented By Gregory Giguashvili

Presented By Gregory Giguashvili GregoryG@ParadigmGeo.com
A Locality-Preserving Cache-Oblivious Dynamic Dictionary By Michael A. Bender, Ziyang Duan, John Iacono, Jing Wu Presented By Gregory Giguashvili

Presented by Gregory Giguashvili
Abstract Simple dictionary designed for a hierarchical memory layouts Cache-oblivious DS has memory performance optimized for all levels of memory hierarchy – no parameterization Locality-preserving dictionary maintains elements of similar key values stored close together 10 December 2018 Presented by Gregory Giguashvili

Abstract Presented a simplification of the cache-oblivious B-tree of Bender, Demain and Farach-Colton Given N elements and B block size Search operations: Insert and delete operations: Scan k data items: Implementation and evaluation of the DS is presented on a simulated memory hierarchy 10 December 2018 Presented by Gregory Giguashvili

Memory Models Two-level memory model Disk Access Model (DAM) consists of internal memory of size M and external disk partitioned into B-sized blocks Focus on particular memory models, thus not portable and not flexible Cache-oblivious memory model Reason about two-level, but prove results for unknown multilevel memory models Parameters B and M are unknown, thus optimized for all levels of memory hierarchy 10 December 2018 Presented by Gregory Giguashvili

B-trees B-tree is the classic external-memory search tree. B-tree supports insert, delete, search and scan on two level memory hierarchy (memory and disk) Balanced tree with fan-out proportional to disk-block size B 10 December 2018 Presented by Gregory Giguashvili

B-trees Formally B-tree is a rooted tree Every node x has the following fields n[x] – number of keys stored in x The keys sorted in non-decreasing order Leaf[x] – Boolean value: TRUE is x is leaf Internal nodes contain pointers to children The keys separate the ranges of keys stored in each sub-tree 10 December 2018 Presented by Gregory Giguashvili

B-trees Formally Every leaf has the same depth (tree’s height) Lower and upper bounds on number of keys per node (minimum degree t2) Every node (other than root) must have at least t-1 keys thus t children Every node can contain at most 2t-1 keys thus 2t children Simplest B-tree occurs when t=2 (2-3-4 tree) In practice, much larger values of t are used 10 December 2018 Presented by Gregory Giguashvili

B-trees Example Search for R M D H Q T X B C F G J K L N P R S V W Y Z 10 December 2018 Presented by Gregory Giguashvili

B-trees Given B-tree with N nodes and B branching factor Tree search: Insert or Delete key: N W Split a node (t=4) N S W P Q R S T U V P Q R T U V 10 December 2018 Presented by Gregory Giguashvili

B-trees Depend critically on block-size B thus optimized only for two levels of memory Theoretically, multilevel B-tree can be created, but it is very complex Improving data layout can optimize B-tree structures (place logically close data physically near each other) Bad performance for sub-optimal B values 10 December 2018 Presented by Gregory Giguashvili

Theory of COM Based on ideal-cache model of Frigo, Leiserson, Prokop and Ramachandran Fully associative cache of size C*B Disk partitioned into B-sized memory blocks Block is fetched from disk to cache on cache-miss Ideal memory block is elected for replacement if cache is full Proved to work with a small constant overhead Used by cache-oblivious algorithms to analyze memory transfers between cache levels 10 December 2018 Presented by Gregory Giguashvili

Data Structure Data array Data and linear number of blank entries Index array Encoding of a tree that indexes the data Search and update operations involve basic manipulations on these arrays Relatively simple to implement 10 December 2018 Presented by Gregory Giguashvili

Related Work Cache-oblivious search tree by Brodal, Fagerberg and Jacob (same complexity) Balanced tree of height logN + O(1) Layout in array of size O(N) Non-data-structure problems Matrix multiplication, transpose, LU decomposition, FFT, sorting Keep N elements ordered given O(N) memory by Itai, Konheim and Rodeh in priority queues 10 December 2018 Presented by Gregory Giguashvili

Data Structure Operations
Dynamic set S with keys from totally ordered universe Insert(x): adds x to S Delete(x): removes x from S Predecessor(x): gets item from S that has the largest key value in S at most x ScanForward(): gets successor of the most recently accessed item in S ScanBackward(): gets predecessor of the most recently accessed item in S 10 December 2018 Presented by Gregory Giguashvili

Data Structure Description
Data array is a packed-memory structure Insertions and deletions require: Scan of k elements requires O(k/B+1) Index array is a static tree structure Each leaf corresponds to an item in data array Updating the index tree does not dominate data insertion cost 10 December 2018 Presented by Gregory Giguashvili

Packed-Memory Maintenance
N totally ordered elements to be sorted in array of size O(N) Element x precedes y iff x < y Any set of k contiguous elements is stored in a contiguous sub-array of size O(k) Scan any set of k contiguous elements uses O(k/B + 1) memory transfers Insert or delete an element uses amortized memory transfers 10 December 2018 Presented by Gregory Giguashvili

When a window of the array becomes too full or empty, the elements are spread evenly within a larger window Window sizes range from O(logN) to O(N) and are powers of two Window sizes are associated with density thresholds (lower LT and upper UT bounds ) When deviation from threshold is discovered, the densities of the windows are adjusted 10 December 2018 Presented by Gregory Giguashvili

Rebalancing Window of size is overflowing if the number of data elements in the region exceeds Window of size is under flowing if the number of data elements in the region is less than Take elements of the window and rearrange them to be spread out evenly 10 December 2018 Presented by Gregory Giguashvili

Insertion of x at location A[j] Examine windows containing A[j] of size for k=loglogN,…,logN Take first window that is not overflowing Insert the element x and rebalance this window Deletion of x from location A[j] Examine windows containing A[j] of size for k=loglogN,…,logN Take first window that is not underflowing Delete the element x and rebalance this window 10 December 2018 Presented by Gregory Giguashvili

Static Tree Maintenance
Cache-oblivious static tree structure due to Prokop Given a complete binary tree, describe mapping from the nodes to positions of an array (van Emde Boas layout) Traverse from root to a leaf in an N node tree in memory transfers (optimal) 10 December 2018 Presented by Gregory Giguashvili

Given a tree with N items and h=log(N+1) Split the tree at the middle A top recursive sub-tree A of height h/2 Bottom recursive sub-trees B1,…,Bk of height h/2 and all sub-trees contain nodes Partition array into k regions, one for each Recursively apply this mapping Stop when the trees have one node 10 December 2018 Presented by Gregory Giguashvili

2 4 5 6 7 8 9 1 3 10 11 12 13 14 15 Van Emde Boas B1 B2 B3 B4 A B1 B2 B3 B4 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Plain Array 10 December 2018 Presented by Gregory Giguashvili

Cache-Oblivious Structure
“Array” refers to the dynamic packed-memory structure storing the data “Tree” refers to the static cache-oblivious tree that serves as an index N data items are stored in “array” of size O(N) (sorted, some blank entries) There are pointers between array position A[i] and leaf Ti for all values of i (A[i] and Ti store the same key value) Internal nodes of T store the maximum of non-blank key values of its children 10 December 2018 Presented by Gregory Giguashvili

Predecessor Operation
Traverse the tree from root to a leaf Similar to standard predecessor search in binary tree Reaching node u, we decide where to branch by comparing the search key to the key of u’s left child 10 December 2018 Presented by Gregory Giguashvili

Predecessor Operation
Theorem 1. The operation predecessor uses block transfers Proof: Search in our structure is similar to a root-to leaf traversal in the “tree”. The process of examining the key value of the current node’s left child at every step might cause additional 10 December 2018 Presented by Gregory Giguashvili

Scan Forward/Backward Operation
Scan forward (backward) the next non-blank item in the “array” from the last item accessed We only scan O(1) elements because of the density constraint of the “array” to find the non-blank element 10 December 2018 Presented by Gregory Giguashvili

Scan Forward/Backward Operation
Theorem 2. A sequence of k scan operations uses block transfers Proof: A sequence of k scan operations only accesses O(k) consecutive elements in the “array” in order. 10 December 2018 Presented by Gregory Giguashvili

Insert/Delete Operation
Insert and delete operations proceed in the same manner – we describe only insert Insert x in three stages: Perform Predecessor(x) to find the location to insert Insert x into the “array” using packed-memory insertion algorithm Update key values in the “tree” 10 December 2018 Presented by Gregory Giguashvili

The first two steps are straightforward Update the key values in the “tree”: Copy the updated key from “array” to “tree” into the corresponding locations Update all the ancestors of the updated leaves in post-order traversal Change key values of the node to reflect the maximum of key values of the children For a given node, the values of its children have been already updated (because of post-order traversal) 10 December 2018 Presented by Gregory Giguashvili

Lemma 1. To perform a post-order traversal on k leaves in the tree and their ancestors requires block transfers. Proof: Consider horizontal stripes formed by the sub-trees, which are smaller than B. We pass stripes on any root-to-leaf path. Number the stripes from the bottom of the tree starting at 1. 10 December 2018 Presented by Gregory Giguashvili

w w-1 w-2 … 2 1 stripes 10 December 2018 Presented by Gregory Giguashvili

Proof (cont.): All items in any stripe are accessible in two memory transfers (MT). We analyze the cost of accessing one of stripe-2 trees and all of its stripe-1 tree children. The size of these trees is in range Since all the stripe-2 tree children are stored consecutively, post order access of k items will cause page faults (provided memory can hold 2 blocks). 10 December 2018 Presented by Gregory Giguashvili

Proof (cont.): We might need one more memory transfer for interleaving accesses. Thus, block transfers are performed (provided memory can hold 4 blocks). In general, access of k consecutive items in post-order mostly consists of level-1 and level-2 sub-trees. There are at most items accessed at stripes 3 and higher. Each of these accesses cause 1 memory transfer (provided memory can hold 5 blocks). 10 December 2018 Presented by Gregory Giguashvili

Proof (cont.): We end up with at most block transfers to access k consecutive items in tree in post-order, given a cache of 5 blocks. 10 December 2018 Presented by Gregory Giguashvili

Theorem 3. The number of block transfers caused by Insert(x) and Delete(x) operations is Proof: The predecessor query costs Insert into packed-memory array costs amortized. Let w be the number of items changed in “array”. Updating the internal nodes in the “tree” uses memory transfers (Lemma 1). 10 December 2018 Presented by Gregory Giguashvili

Proof (cont.): Since is asymptotically the same as the number of block transfers during insertion into “array”, it may be restated as amortized cost of insertion into the packed-memory structure. Therefore, the operation takes: 10 December 2018 Presented by Gregory Giguashvili

Simulation Results How the block size B and cache size M affect DS compared to B-tree??? Up to 1,000,000 elements inserted Each element is 32-bit integer If array becomes too full, recopy elements to a larger array 10 December 2018 Presented by Gregory Giguashvili

Simulation Input Patterns
Insert at head Insert in the beginning of the array Worst case behavior Random insert Insert before a random element in the array Assign random 32-bit key to new element Bulk insert Insert a sequence of elements at random position 10 December 2018 Presented by Gregory Giguashvili

Experiments Description
Packed-memory structure Amortized number of moved elements per insertions Different thresholds and densities considered Memory simulator Model blocks in memory and on disk LRU strategy for block replacement Measure page faults by packed-memory and index structures separately 10 December 2018 Presented by Gregory Giguashvili

Scans and Moves Count number of elements moved during rebalance operation Density parameters of 50% up to 90% Average number of moves during head insertions ~350 per 1,000,000 elements Random insertions have a small constant overhead Number of moves increases with the bulk size 10 December 2018 Presented by Gregory Giguashvili

Page Faults Packed-memory structure (insert at head) More sensitive to B than to N Almost linear dependence on size of B Gets worse if rebalance window is less than cache size Packed-memory structure (random insert) Close to the worst case (1 fault per insert) Packed-memory structure (bulk insert) Insert cost is amortized by the number of elements inserted 10 December 2018 Presented by Gregory Giguashvili

Page Faults Entire structure (searching index+scanning the array) Consider bulk sizes 1,10,100,…, Worst case is the random inserts Increasing bulk size gives order of magnitude performance increase Larger bulk sizes hurt performance while rebalancing the windows Performance is never worse than random insertions 10 December 2018 Presented by Gregory Giguashvili

B-tree Simulations Data entries are 32-bit integers Each node contains at most B data and B+1 branches Tested with different B and bulk sizes Insert at head is the best case Keeping B close to page size improves performance Searching a random key in tree of 1,000,000 elements with page size of 1024 bytes, gives average page fault rate of 3.77 (against 3.69 in CODS). 10 December 2018 Presented by Gregory Giguashvili

Conclusions Advantages over standard B-tree Cache-oblivious Locality preserving Two static arrays Worst-case performance CODS is as good as of B-tree for typical B and M Optimized over multiple level memory hierarchies 10 December 2018 Presented by Gregory Giguashvili

The End 10 December 2018 Presented by Gregory Giguashvili

Presented By Gregory Giguashvili

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