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A.2 Time and Space Considerations

This section presents a very approximate analysis and comparison of the time and space requirements for a variety of common operations on different specific collections and collection families. We've presented the information as a set of tables that lists the operation, the time cost and the space cost. Any applicable comments appear at the bottom of the table. A key to the abbreviations used in the tables appears at the bottom of each page.

As you read these analyses, keep in mind that various processors, operating systems, compilation optimizations, and many other factors will affect the exact values. The point of these tables is to provide you with some idea of how the behaviors of the various collections will compare, all other things being equal. For more details on algorithm complexity, refer to Knuth, Sedgewick, or any number of other books.

Because many of the Tools.h++ collections have essentially similar interfaces, it is easy to experiment and discover what effect a different choice of collection will have on your program.

For each of the following tables:

Whenever an allocation is mentioned, you should be aware that memory allocation policies differ radically among various implementations. However, it is generally true that a heap allocation (or deallocation) that translates to a system call is more expensive than most of the other constant costs. "Amortized" cost is averaged over the life of the collection. Any individual action may have a higher or lower cost than the amortized value.

Key to the comparison tables

N
M
t
i
p
C
count of items
count of space for items
sizeof (item)
sizeof (int)
sizeof (void*)
a constant

A.2.1 RWGVector, RWGBitVec, RWTBitVec<size>, RWTPtrVector, and RWTValVector

operation
time cost
space cost
Find (average item)
N/2
0
Change/replace item
C
0
Container overhead
  (Mt+2i) + 0
Comments
Simple wrapper around an array of T (except bitvec: array of byte)
Resize only if told to.

A.2.2 Singly Linked Lists

operation
time cost;
space cost
Insert at either end
C
t + p
Insert in middle
C (assumes that you have an iterator in position)
t + p
Find (average item)
N/2
0
Change/replace item
C
0
Remove first
C
t + p (recovered)
Remove last
C
t + p (recovered)
Remove in middle
C (assumes that you have an iterator in position)
t + p (recovered)
Container overhead
  (2p+i) + N(t+p)
Comments
Allocation with each insert
Iterators go forward only
Grows or shrinks with each item.
Smaller than doubly-linked list

A.2.3 Doubly Linked Lists

operation
time cost;
space cost
Insert at either end
C
t + 2p
Insert in middle
C (assumes that you have an iterator in position)
t + 2p
Find (average item)
N/2
0
Change/replace item
C
0
Remove first
C
t + 2p (recovered)
Remove last
C
t + 2p (recovered)
Remove in middle
C (assumes that you have an iterator in position)
t + 2p (recovered)
Container overhead
  (2p+i) + N(t+2p)
Comments
Allocation with each insert
Iterate in either direction
Grows or shrinks with each item
Larger than Slist

A.2.4 Ordered Vectors

operation
time cost;
space cost
Insert at the end
C (amortized)
t (amortized)
Insert in middle
N/2
t (amortized)
Find (average item)
N/2
0
Change/replace item
C
0
Remove first
N
0
Remove last
C
0
Remove in middle
N/2
0
Container overhead
  (Mt+ p + 2i) + 0
Comments
No iterators (use size_t index)
Allocation only when the vector grows.
Expands as needed by adding space for many entries at once. Shrinks only via resize()

A.2.5 Sorted Vectors

operation
time cost;
space cost
Insert
logN + N/2 (average)
t (amortized)
Find (average item)
logN
0
Change/replace item
N
0
Remove first
N
0
Remove last
C
0
Remove in middle
N/2
0
Container overhead
  (Mt + p + 2i) + 0
Comments
Insertion happens based on sort order.
No iterators (use size_t index)
replace requires remove/add sequence to maintain sorting
Allocation only when the vector grows.
Expands as needed by adding space for many entries at once. Shrinks only via resize()

A.2.6 Stacks and Queues

operation
time cost;
space cost
Insert at end
C
t + p
Remove (pop)
C
t + p (recovered)
Container overhead
  (2p+i) + N(t+p)
Comments:
Implemented as singly -linked list.
Templatized version allows choice of container: time and space costs will reflect that choice.
 

A.2.7 Deques

operation
time cost;
space cost
Insert at end
C
t (amortized)
Find (average item)
N/2
0
Change/replace item
C
0
Remove first
C
t (amortized, recovered)
Remove last
C
t (amortized, recovered)
Remove in middle
N/2
t (amortized, recovered)
Container overhead
  (Mt + p + i) + 0
Comments
Implemented as circular queue in an array.
Allocation only when collection grows
Expands and shrinks as needed, caching extra expansion room with each increase in size

A.2.8 Binary Tree

operation
time cost;
space cost
Insert
logN+C
2p+t
Find (average item)
logN
0
Change/replace item
2(logN + C)
0
Remove first
logN + C
2p+t (recovered)
Remove last
logN + C
2p+t (recovered)
Remove in middle
logN + C
2p+t (recovered)
Container overhead
  (p+i) + N(2p+t)
Comments
Insertion happens based on sort order.
Allocation with each insert
Replace requires remove/add sequence to maintain order
Does not automatically remain balanced. Numbers above assume a balanced tree.
Costs same as doubly linked list per item

A.2.9 (multi)map and (multi)set family

operation
time cost;
space cost
Insert
logN+C
2p+t
Find (average item)
logN
0
Change/replace item
2(logN+C) or C
0
Remove first
logN (worst case)
2p+t (recovered)
Remove last
logN (worst case)
2p+t (recovered)
Remove in middle
logN (worst case)
2p+t (recovered)
Container overhead
re-balance may occur at each insert or remove
(3p+i) + N(2p+t)
Comments
Insertion happens based on sort order.
Allocation with each insert
Replace for sets requires remove/insert. For maps the value is copied in place.
implemented as balanced (red-black) binary tree.
 

A.2.10 RWBTree, RWBTreeDictionary

RWBTreeOnDisk has complexity similar to RWBTreeDictionary, but the time overhead is much greater since "following linked nodes" becomes "disk seek;" and the size overhead has a much greater impact on disk storage than on core memory.

operation
time cost;
space cost
Insert
logN+C
2p + t + small (fully amortized)
Find (average item)
logN
0
Change/replace item
2logN+2 or C
0
Remove first
2logN(log2(ORDER))+C
(worst case)
2p+t (recovered)
Remove last
2logN(log2(ORDER))+C
(worst case)
2p+t (recovered)
Remove in middle
2logN(log2(ORDER))+C (worst case)
2p+t (recovered)
Container overhead
Re-balance may occur at each insert or remove. However it will happen less often than for a balanced binary tree.
This depends on how fully the nodes are packed. Each node costs ORDER(2p+t+1)+i and there will be no more than 2N/ORDER, and no fewer than min(N/ORDER,1) nodes. Inserting presorted items will tend to maximize the size.
Sedgewick says the size is close to 1.44 N/ORDER for random data
Comments
Insertion based on sort order.
The logarithm is approximately base ORDER which is the splay of the b-tree. (In fact the base is between ORDER and 2ORDER depending on the actual loading of the b-tree)
Replace for b-tree requires remove then insert. For btreedictionary the value is copied in place
 

A.2.11 Hash-based Collections

RWSet and RWIdentitySet as well as collections with "Hash" in their names.

operation
time cost;
space cost
Insert
C
p+t
Find (average item)
C
0
Change/replace item
2C
0
Remove
C
p+t (recovered)
Container overhead
  ((M+2)p+i) + N(p+t) (1) (Mp+(2p+i)b_used) + N(p+t) (2)
1: standard compliant version 2: b_used is "number of used slots" for the "V6.1" hashed collections
Comments
Insertion happens based on the hashing function.
Constant time costs assume that the items are well scattered in the hash slots. Worst case is linear in the number of items per slot.
Replace for dictionary or map: The new value is copied in place. Otherwise, requires remove then insert.
Does not automatically resize.
We recommend that the number of items be between one half and double the number of slots for most uses.


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