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.

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.

Table 22 – Time and space requirements for 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.

Table 23 – Time and space requirements for 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

Table 24 – Time and space requirements for 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.

Table 25 – Time and space requirements for 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	Allocation only when the vector grows.	Expands as needed by adding space for many entries at once. Shrinks only via resize()

Table 26 – Time and space requirements for 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 for non-templatized classes (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()

Table 27 – Time and space requirements for 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.

Table 28 – Time and space requirements for 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

Table 29 – Time and space requirements for 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

Table 30 – Time and space requirements for (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.

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.

Table 31 – Time and space requirements for RWBTree and RWBTreeDictionary
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

Table 32 – Time and space requirements for hashed-based collections
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.