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Partition representations: Difference between revisions
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Partition representations (view source)
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== Definitions == | == Definitions == | ||
In this page, a '''partition''' of a [[vector]] is defined to be a [[nested]] or [[box]]ed vector containing vectors, such that the [[Raze]] (or equivalently <source lang=apl inline>{⊃⍪/⍵}</source> in [[Nested array model|nested]] APLs) of the partition intolerantly [[match]]es the original vector. A partition contains all the original [[element]]s of the vector, in the same order, but at one greater [[depth]]. | In this page, a '''partition''' of a [[vector]] is defined to be a non-[[empty]] [[nested]] or [[box]]ed vector containing vectors, such that the [[Raze]] (or equivalently <source lang=apl inline>{⊃⍪/⍵}</source> in [[Nested array model|nested]] APLs) of the partition intolerantly [[match]]es the original vector. A partition contains all the original [[element]]s of the vector, in the same order, but at one greater [[depth]]. | ||
The vectors contained in a partition are called '''divisions''', and the boundaries between them are called '''dividers'''. Although the English word "partition" can be used for either of these, the ambiguity of using one word for three different objects could be confusing. Only boundaries between divisions are called dividers: we do not name the two outermost edges, which must exist in any partition. | The vectors contained in a partition are called '''divisions''', and the boundaries between them are called '''dividers'''. Although the English word "partition" can be used for either of these, the ambiguity of using one word for three different objects could be confusing. Only boundaries between divisions are called dividers: we do not name the two outermost edges, which must exist in any partition. | ||
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== Representation using divider locations == | == Representation using divider locations == | ||
Existing APLs rarely define partitioning functions using properties of the resulting divisions. Instead, most partition functions use a vector of the same length as the partitioned vector to control how it is partitioned. For example, [[Partitioned Enclose]] starts a new | Existing APLs rarely define partitioning functions using properties of the resulting divisions. Instead, most partition functions use a vector of the same length as the partitioned vector to control how it is partitioned. For example, [[Partitioned Enclose]] starts a new division whenever a 1 is encountered in the [[Boolean]] left argument: | ||
<source lang=apl> | <source lang=apl> | ||
1 0 1 1 0 0 1 ⊂ 'abcdefg' | 1 0 1 1 0 0 1 ⊂ 'abcdefg' | ||
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{{Works in|[[Dyalog APL]]}} | {{Works in|[[Dyalog APL]]}} | ||
This style of partition function is slightly harder to understand and implement, but does have advantages: | This style of partition function is slightly harder to understand and implement, but does have advantages: | ||
* The left argument can sometimes be obtained directly from the right argument. For example, a | * The left argument can sometimes be obtained directly from the right argument. For example, a division might be started whenever a space is encountered. | ||
* Left arguments can be merged using arithmetic. For two [[Partitioned Enclose]] arguments <source lang=apl inline>a</source> and <source lang=apl inline>b</source>, the [[Maximum]] <source lang=apl inline>a⌈b</source> gives a common [[wikipedia:Partition of a set#Refinement of partitions|refinement]] of the two corresponding partitions. | * Left arguments can be merged using arithmetic. For two [[Partitioned Enclose]] arguments <source lang=apl inline>a</source> and <source lang=apl inline>b</source>, the [[Maximum]] <source lang=apl inline>a⌈b</source> gives a common [[wikipedia:Partition of a set#Refinement of partitions|refinement]] of the two corresponding partitions. | ||
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└──┴────┴─┘ | └──┴────┴─┘ | ||
</source> | </source> | ||
A more rigorous approach is to allow the left argument to specify the [[index]] of the division to which the corresponding element belongs. In order for this to produce a partition, the left argument must be a vector of non-decreasing non-negative integers (as with the division endpoint representation). Then for a left argument <source lang=apl inline>l</source> the number of | A more rigorous approach is to allow the left argument to specify the [[index]] of the division to which the corresponding element belongs. In order for this to produce a partition, the left argument must be a vector of non-decreasing non-negative integers (as with the division endpoint representation). Then for a left argument <source lang=apl inline>l</source> the number of divisions is <source lang=apl inline>⊃⌽l</source>. BQN's [[Group (BQN)|Group]] uses an extension of this representation. | ||
<source lang=apl> | <source lang=apl> | ||
1 1 3 3 3 3 6 part 'abcdefg' | 1 1 3 3 3 3 6 part 'abcdefg' | ||
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└┴──┴┴────┴┴┴─┘ | └┴──┴┴────┴┴┴─┘ | ||
</source> | </source> | ||
As with the Partition-based representation, we must allow an additional element in the left argument after the end of the right argument in order to encode empty | As with the Partition-based representation, we must allow an additional element in the left argument after the end of the right argument in order to encode empty divisions after the end of the data. | ||
This definition is nearly identical to the one used in [[Partitioned Enclose]] when the divider counts are [[Boolean]]. The only difference is in the first element: while in Partitioned Enclose a first element of 0 indicates that no division is started, and elements before the first 1 are excluded from the final | This definition is nearly identical to the one used in [[Partitioned Enclose]] when the divider counts are [[Boolean]]. The only difference is in the first element: while in Partitioned Enclose a first element of 0 indicates that no division is started, and elements before the first 1 are excluded from the final division, in the version here, a first element of 0 is like a 1 in Partitioned Enclose indicating that a division is started, while an initial 1 indicates that an empty division precedes the first non-empty division. The number used in the representation here is one higher than the one used by Partitioned Enclose. This encoding better represents complete partitions of the right argument because every vector on non-negative integers corresponds to a partition. In Partitioned Enclose vectors beginning with 0 correspond to incomplete partitions where some initial elements are not included. However, the partitioning used in Partitioned Enclose can still be produced by [[drop]]ping the first division. | ||
== Unification == | == Unification == | ||
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Because it interleaves elements from a vector with dividers, partitioning is closely related to the [[Mesh]] operation. In fact, a partition can be used to produce a mesh by prepending an element to each division and [[Raze]]ing: | Because it interleaves elements from a vector with dividers, partitioning is closely related to the [[Mesh]] operation. In fact, a partition can be used to produce a mesh by prepending an element to each division and [[Raze]]ing: | ||
<source lang=apl> | <source lang=apl> | ||
⎕←P | |||
┌┬──┬┬────┬┬┬─┐ | ┌┬──┬┬────┬┬┬─┐ | ||
││ab││cdef│││g│ | ││ab││cdef│││g│ | ||
| Line 145: | Line 145: | ||
</source> | </source> | ||
In the above functions, the only difference in code is exchanging 0 and 1, and the only difference in output is an extra trailing 0 when computed from the division lengths. The mesh representation makes the partition representation duality especially clear, as the dual representation can be obtained simply by [[Not|negating]] the mesh vector. | In the above functions, the only difference in code is exchanging 0 and 1, and the only difference in output is an extra trailing 0 when computed from the division lengths. The mesh representation makes the partition representation duality especially clear, as the dual representation can be obtained simply by [[Not|negating]] the mesh vector. | ||
[[Category:Essays]] | |||