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Graph and tree search algorithms
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In computer science, B* (pronounced 'B star') is a best-firstgraph search algorithm that finds the least-cost path from a given initial node to any goal node (out of one or more possible goals). First published by Hans Berliner in 1979, it is related to the A* search algorithm.

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2Details
- 2.4Expansion

Summary[edit]

The algorithm stores intervals for nodes of the tree as opposed to single point-valued estimates. Then, leaf nodes of the tree can be searched until one of the top level nodes has an interval which is clearly 'best.'

Details[edit]

Interval evaluations rather than estimates[edit]

Leaf nodes of a B*-tree are given evaluations that are intervals rather than single numbers. The interval is supposed to contain the true value of that node. If all intervals attached to leaf nodes satisfy this property, then B* will identify an optimal path to the goal state.

Backup process[edit]

To back up the intervals within the tree, a parent's upper bound is set to the maximum of the upper bounds of the children. A parent's lower bound is set to the maximum of the lower bound of the children. Note that different children might supply these bounds.

Termination of search[edit]

B* systematically expands nodes in order to create 'separation,' which occurs when the lower bound of a direct child of the root is at least as large as the upper bound of any other direct child of the root. A tree that creates separation at the root contains a proof that the best child is at least as good as any other child.

In practice, complex searches might not terminate within practical resource limits. So the algorithm is normally augmented with artificial termination criteria such as time or memory limits. When an artificial limit is hit, then you must make a heuristic judgment about which move to select. Normally, the tree would supply you with extensive evidence, like the intervals of root nodes.

Expansion[edit]

B* is a best-first process, which means that it is very efficient to traverse the tree, repeatedly descending to find a leaf to expand. This section describes how to choose the node to expand. (Note: Whether or not the tree is memory-resident, is a function of the overall implementation efficiency, including how it may be mapped and/or managed via real or virtual memory.)

At the root of the tree, the algorithm applies one of two strategies, called prove-best and disprove-rest. In the prove-best strategy, the algorithm selects the node associated with the highest upper bound. The hope is that expanding that node will raise its lower bound higher than any other node's upper bound.

The disprove-rest strategy selects the child of the root that has the second-highest upper bound. The hope is that by expanding that node you might be able to reduce the upper bound to less than the lower bound of the best child.

Strategy selection[edit]

Note that applying the disprove-rest strategy is pointless until the lower bound of the child node that has the highest upper bound is the highest among all lower bounds.

The original algorithm description did not give any further guidance on which strategy to select. There are several reasonable alternatives, such as expanding the choice that has the smaller tree.

Strategy selection at non-root nodes[edit]

Once a child of the root has been selected (using prove-best or disprove-best) then the algorithm descends to a leaf node by repeatedly selecting the child that has the highest upper bound.

When a leaf node is reached, the algorithm generates all successor nodes and assigns intervals to them using the evaluation function. Then the intervals of all nodes have to be backed up using the backup operation.

When transpositions are possible, then the back-up operation might need to alter the values of nodes that did not lie on the selection path. In this case, the algorithm needs pointers from children to all parents so that changes can be propagated. Note that propagation can cease when a backup operation does not change the interval associated with a node.

Robustness[edit]

If intervals are incorrect (in the sense that the game-theoretic value of the node is not contained within the interval), then B* might not be able to identify the correct path. However, the algorithm is fairly robust to errors in practice.

The Maven (Scrabble) program has an innovation that improves the robustness of B* when evaluation errors are possible. If a search terminates due to separation then Maven restarts the search after widening all of the evaluation intervals by a small amount. This policy progressively widens the tree, eventually erasing all errors.

Extension to two-player games[edit]

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The B* algorithm applies to two-player deterministic zero-sum games. In fact, the only change is to interpret 'best' with respect to the side moving in that node. So you would take the maximum if your side is moving, and the minimum if the opponent is moving. Equivalently, you can represent all intervals from the perspective of the side to move, and then negate the values during the back-up operation.

Applications[edit]

Andrew Palay applied B* to chess. Endpoint evaluations were assigned by performing null-move searches. There is no report of how well this system performed compared to alpha-beta pruning search engines running on the same hardware.

The Maven (Scrabble) program applied B* search to endgames. Endpoint evaluations were assigned using a heuristic planning system.

The B* search algorithm has been used to compute optimal strategy in a sum game of a set of combinatorial games.

References[edit]

Berliner, Hans (1979). 'The B* Tree Search Algorithm. A Best-First Proof Procedure'. Artificial Intelligence. 12 (1): 23–40. doi:10.1016/0004-3702(79)90003-1.
Russell, S. J.; Norvig, P. (2003). Artificial Intelligence: A Modern Approach. Upper Saddle River, N.J.: Prentice Hall. p. 188. ISBN0-13-790395-2.
Sheppard, Brian (2002). 'World-championship-caliber Scrabble'. Artificial Intelligence. 134 (1–2): 241–275. doi:10.1016/S0004-3702(01)00166-7.

Retrieved from 'https://en.wikipedia.org/w/index.php?title=B*&oldid=866699604'

1Translingual
- 1.1Etymology 1
  - 1.1.1Letter
- 1.2Etymology 2
2English
- 2.2Etymology 1
  - 2.2.1Letter
- 2.3Etymology 2
- 2.4Etymology 3
  - 2.4.1Noun
  - 2.4.3Symbol
3Afar
4American Sign Language
5Azerbaijani
- 5.1Letter
6Blagar
7Catalan
- 7.2Letter
9Czech
- 9.2Letter
10Dutch
11Esperanto
- 11.2Letter
12Finnish
- 12.1Letter
13German
- 13.2Letter
- 13.3Noun
14German Sign Language
15Ido
- 15.2Letter
16Italian
17Latvian
- 17.3Letter
18Malay
- 18.2Letter
19Maltese
20Portuguese
- 20.1Letter
21Romanian
- 21.2Letter
22Saanich
- 22.2Letter
23Skolt Sami
24Slovene
- 24.1Letter
25Somali
- 25.2Letter
26Spanish
- 26.1Letter
27Turkish
- 27.1Letter
28Vietnamese
- 28.2Letter
29Zulu
- 29.1Letter

Translingual[edit]

B	U+0042, B LATIN CAPITAL LETTER B

← A
[U+0041]

Basic Latin

C →
[U+0043]

Ｂ

U+FF22, Ｂ

← Ａ
[U+FF21]

Halfwidth and Fullwidth Forms

Ｃ →
[U+FF23]

Etymology 1[edit]

From the Ancient Greek letter Β(B, “Beta”).

Letter[edit]

B (lower caseb)

The second letter of the basic modern Latin alphabet.

Usage notes[edit]

The letter B (usually its lower case form b) is used in several romanization systems of non-Latin scripts to represent the bilabial plosive or stop, usually voiced (/b/).

Etymology 2[edit]

(blood type): from antigen B
(symbol for boron): abbreviation of boron
(hexadecimal 11): The eleventh item from the sequence {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

Symbol[edit]

A blood type that has a specific antigen that aggravates the immune response in people with type A antigen in their blood. They can receive blood from type B or type O, but cannot receive blood from AB or A.
(chemistry) The chemical element boron.
Eleven, especially used in the base more than ten, such as duodecimal, hexadecimal, vigesimal and so on.
The equivalent numbers of decimal 1611 are duodecimal B23, hexadecimal 64B, vigesimal 40B and so on.
(biochemistry)As an IUPAC code for amino acids,aspartic acid or asparagine.
(physics) A magnetic flux density.

Gallery[edit]

Letter styles
Capital and lowercase versions of B, in normal and italic type
Uppercase and lowercase B in Fraktur
B in uncial script

English[edit]

Pronunciation[edit]

(phoneme)IPA^(key): /b/
(letter name)IPA^(key): /biː/
Audio (UK)
Audio (AU)
Rhymes: -iː

Etymology 1[edit]

Old English letter B, from 7th century replacement by LatinB of the Anglo-Saxon Futhorc letter ᛒ(B, “beorc”).

Letter[edit]

B (upper case, lower caseb, pluralBsorB's)

The second letter of the English alphabet, called bee and written in the Latin script.
- 1945 August 17, George Orwell [pseudonym; Eric Arthur Blair], chapter 3, in Animal Farm: A Fairy Story, London: Secker & Warburg, :
  Boxer could not get beyond the letter D. He would trace out A, B, C, D, in the dust with his great hoof […]

Usage notes[edit]

In Modern English, the letter B is sometimes silent, e.g. in climb, and debt.

Apostrophes are sometimes used to form plurals of single upper case letters, especially at the start of a sentence:

A’s are difficult to achieve in this class.

Usually, though, the apostrophe is avoided when there is no risk of confusion:

She gets straight As in school.

Number[edit]

B (upper case, lower caseb)

The ordinal number second, derived from this letter of the English alphabet, called bee and written in the Latin script.

Etymology 2[edit]

(cricket, balls): abbreviation of balls
(billion): abbreviation of billion
(black): abbreviation of black

Noun[edit]

(cricket) the number of balls faced by a batsman
The quantity one billion (1,000,000,000), usually used to signify a sum of money
$2.8 B — “two point eight billion dollars” (i.e., $2,800,000,000)
(Britain) The grade of pencil that makes darker marks than grade HB but lighter marks than grade 2B; a pencil with soft lead.
(chess)Bishop

Etymology 3[edit]

(personality type): from contrast with the letter ‘A’ and its corresponding personality type
(academic grade): from the position of the letter ‘B’ in the English alphabet

Noun[edit]

B (pluralBs)

A personality type of someone who is relaxed and easygoing and able to engage in leisure activities without worrying about work.
An academic grade, better than a C and worse than an A
Signifies a second-tier or second class of a given commodity, group, or category, as in B-movie, B-list, etc.
The television studio revived some of it's B characters for the spinoff.
Alternate or secondary part, as the back side of a phonograph record. Contrasted with A, which is the primary part.
(music) The seventh note in the C majorscale.

Translations[edit]

Czech: H(cs)
Faroese: Hn
Finnish: H(fi)

French: si(fr)m
German: H(de)n

Usage notes[edit]

Apostrophes are sometimes used to form plurals of single upper case letters, especially at the start of a sentence:

A’s are difficult to achieve in this class.

Usually, though, the apostrophe is avoided when there is no risk of confusion:

She gets straight As in school.

Symbol[edit]

(computing)byte

Derived terms[edit]

(computing):Bps, B/s
(computing):kB, MB, GB, TB, PB, EB

Translations[edit]

French: o(fr)(octet)

Proper noun[edit]

(computer languages) A programming language from which C is derived.

Afar[edit]

Letter[edit]

The second letter in the Afar alphabet.

American Sign Language[edit]

Alternative forms[edit]

(thumb unopposed and flat against the radial edge of the palm, used in rapid fingerspelling of words like B-A-B-Y)FlatB

Etymology[edit]

From French Sign Language sign for the letter B.

Production[edit]

This one-handed ASL sign is produced as follows:

Posture the dominant hand in the “B” handshape .

Letter[edit]

(transliteration needed) (StokoeB)

The letter B

Azerbaijani[edit]

Letter[edit]

Bupper case (lower caseb)

The second letter of the Azerbaijani alphabet, written in the Latin script.

Blagar[edit]

Letter[edit]

The second letter of the Blagar alphabet, written in the Latin script.

Catalan[edit]

Pronunciation[edit]

IPA^(key): /be/

Letter[edit]

B (upper case, lower caseb)

The second letter of the Catalan alphabet, written in the Latin script.

Chinese[edit]

For pronunciation and definitions of B – see 屄‎ (“vulva; cunt; etc.”).
(This term, B, is a variant form of 屄‎.)

Czech[edit]

Pronunciation[edit]

IPA^(key): /beː/

Letter[edit]

B(the 2nd letter in the Czech alphabet)

Noun[edit]

(music)Bflat

Dutch[edit]

Pronunciation[edit]

IPA^(key): /beː/
Audio

Letter[edit]

B (upper case, lower caseb)

The second letter of the Dutch alphabet, written in the Latin script.

Esperanto[edit]

Pronunciation[edit]

(letter name):IPA^(key): /bo/
(phoneme):IPA^(key): /b/
Audio

Letter[edit]

B (upper case, lower caseb)

The second letter of the Esperanto alphabet, called bo and written in the Latin script.

Finnish[edit]

Letter[edit]

B (upper case, lower caseb)

The second letter of the Finnish alphabet, called bee and written in the Latin script.

Usage notes[edit]

Used only in loanwords.

Noun[edit]

Abbreviation of lubenter approbatur.

German[edit]

Pronunciation[edit]

IPA^(key): /beː/
Audio

Letter[edit]

B (upper case, lower caseb)

The second letter of the German alphabet, called be and written in the Latin script.

Noun[edit]

Bn (genitiveB, no plural)

(music)B-flat

Declension[edit]

singular
indef.	def.	noun
nominative	ein	das	B
genitive	eines	des	B
dative	einem	dem	B
accusative	ein	das	B

German Sign Language[edit]

Etymology[edit]

Related to the French Sign Language sign for 'B'.

Production[edit]

This one-handed GSL sign is produced as follows:

Posture the dominant hand in the “B” shape.

Letter[edit]

(transliteration needed)

the letter B

Ido[edit]

Pronunciation[edit]

IPA^(key): /b/

Letter[edit]

B (lower caseb)

The second letter of the Ido alphabet, written in the Latin script.

Italian[edit]

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Italian Wikipedia has an article on:

Wikipedia ^it

Pronunciation[edit]

(letter name)IPA^(key): /bi/
(phoneme)IPA^(key): /b/

Letter[edit]

Bm or f (lower caseb)

The second letter of the Italian alphabet, called bi and written in the Latin script.