Making B+-Trees Cache Conscious in Main Memory Author:Jun Rao, Kenneth A. Ross Members: Iris Zhang, Grace Yung, Kara Kwon, Jessica Wong

Outline 1. Introduction 2. Related Work 3. Cache Sensitive B+-Trees 4. Conclusion

Motivation Significant portion of execution time: second level data cache misses first level instruction cache misses System Hierarchy

Motivation (Cont’d) 2. CPU speeds have been increasing at a much faster rate than memory speeds Conclusion: improving cache behavior is going to be an imperative task in main memory data processing Resolution: using memory index structure

Cache Memories Cache memories are small fast static RAM memories that improve performance by holding recently referenced data. Parameter: Capacity Block Size (cache line) Associativity Memory reference: Hit Miss

Cache Optimization on Index Structures—B+-Trees Height-balanced tree Minimum 50% occupancy (except for root). Each node contains d <= m <= 2d entries. The parameter d is called the order of the tree. (n=2d) Each node is 1 cache line (cache-line based) Full pointer B+-Tree (n =2)

Cache Optimization on Index Structures—CSS-Trees Similar as B+-tree Eliminating child pointers Storing child nodes in a fixed sized array. Nodes are numbered & stored level by level, left to right. Position of child node can be calculated via arithmetic. No pointer CSS-Tree

Comparison between B+-Trees and CSS-Trees Cache Line Size=12 bytes, Key Size=Pointer Size=4 bytes Search key =3 B+-Tree CSS-Tree

Comparison between B+-Trees and CSS-Trees(cont’d) full pointer more cache access and more cache misses efficient for updating operation, e.g. insertion and deletion CSS tree no pointer fewer cache access and fewer cache misses acceptable for static data updated in batches Pointer elimination is important in cache optimization, but removing pointer completely introduces some restriction, so we use partial elimination Conclusion: partial pointer elimination

Cache Sensitive B+-Trees Cache Sensitive B+-Trees with One Child Pointer Segmented CSB+-Trees Full CSB+-Trees

Cache Sensitive B+-Trees with One Pointer Similar as B+-tree All the child nodes of any given node are put into a node group with one pointer Nodes within a node group are stored continuously and can be accessed using an offset to the first node in the group

Cache Sensitive B+-Trees with One Pointer (cont’d) Cache misses are reduced because a cache line can hold more keys than B+-Trees and can satisfy one more level comparison. CSB+-Tree can support incremental updates in a way similar to B+-Tree Cache Line Size=64 bytes, Key Size=Pointer Size=4 bytes B+-Tree: 7 keys per node CSB+-Tree: 14 keys per node

Operations on CSB+-Tree—Bulkload 22| 7| 30| 3| 13|19 25| 33| 2|3 5|7 12|13 16|19 20|22 24|25 27|30 31|33 36|39

Operations on CSB+-Tree— Insertion Search the leaf node n to insert the new entry If n is not full, insert the new entry in the appropriate place Otherwise, split n. Let p be n’ parent node, f be the first-child pointer in p and g be the node-group pointed by f If p is not full, copy g to g' in which n is split in two nodes. Let f point to g' If p is full, copy half g to g'. Let f point to g'. Split the node-group of p according to step a

Operations on CSB+-Tree— Insertion (cont’d) 22| key = 34 7| 30| 3| 13|19 25| 33| 2|3 5|7 12|13 16|19 20|22 24|25 27|30 31|33 36|39 a CSB+-Tree of Order 1

Operations on CSB+-Tree— Insertion (cont’d) 22| key = 34 7| 30| 3| 13|19 25| 33|36 2|3 5|7 12|13 16|19 20|22 24|25 27|30 31|33 34|36 39|

Operations on CSB+-Tree—Search Determine the rightmost key K in the node that is smaller than the search key Get the address of the child node Goto first step until find the search key or there is no other node can be checked Search method in a node basic approach uniform approach variable approach

Segmented Cache Sensitive B+-Trees Problem: it’s time consuming to split a node group Resolution:SCSB+-Tree method: divide node group into two segments with one child pointer per segment result: better split performance, but worse search

Full CSB+-Tree Motivation: reduce the split cost Method: Result: pre-allocate space for a full node group shift part of the node group along by one node when a node split Result: reduce the split cost, but increase the space complexity

Conclusion CSB+-Trees are more cache conscious than B+-Tree because of partial pointer elimination CSB+-Trees support efficient incremental updates, but CSS-Trees do not Partial pointer elimination is a general technique which can be applied to other memory structures