[SS] 16Apr2004: I wrote these two functions some time ago because someone posted in comp.lang.python a related question, i.e. just for fun, but probably this can be of real use. The functions are useful to compute the permutations of a set, actually you in theory only need [[permutations]] - that returns a list of permutations of the input set (also a Tcl list), but it's very memory-consuming to generate the permutations and then use [foreach] against it, so there is another function called [[foreach-permutation]] that runs all the permutations of the set and run a script against every one. The code is Copyright(C) 2004 Salvatore Sanfilippo, and is under the Tcl 8.4 license. '''Examples''' % !source source antilib.tcl % permutations {a b c} {a b c} {a c b} {b a c} {b c a} {c a b} {c b a} % foreach-permutation p {a b c} {puts \$p} a b c a c b b a c b c a c a b c b a % '''Code''' # Return a list with all the permutations of elements in list 'items'. # # Example: permutations {a b} > {{a b} {b a}} proc permutations items { set l [llength \$items] if {[llength \$items] < 2} { return \$items } else { for {set j 0} {\$j < \$l} {incr j} { foreach subcomb [permutations [lreplace \$items \$j \$j]] { lappend res [concat [lindex \$items \$j] \$subcomb] } } return \$res } } # Like foreach but call 'body' for every permutation of the elements # in the list 'items', setting the variable 'var' to the permutation. # # Example: foreach-permutation x {a b} {puts \$x} # Will output: # a b # b a proc foreach-permutation {var items body} { set l [llength \$items] if {\$l < 2} { uplevel [list set \$var [lrange \$items 0 0]] uplevel \$body } else { for {set j 0} {\$j < \$l} {incr j} { foreach-permutation subcomb [lreplace \$items \$j \$j] { uplevel [list set \$var [concat [lrange \$items \$j \$j] \$subcomb]] uplevel \$body } } } } ---- [RS]: Here's my take, a tiny hot summer day balcony fun project: proc permute {list {prefix ""}} { if ![llength \$list] {return [list \$prefix]} set res {} foreach e \$list { lappend res [permute [l- \$list \$e] [concat \$prefix \$e]] } join \$res } proc l- {list e} { set pos [lsearch \$list \$e] lreplace \$list \$pos \$pos } [Lars H]: RS's '''permute''', tidied up: proc permute {list {prefix ""}} { if {![llength \$list]} then {return [list \$prefix]} set res [list] set n 0 foreach e \$list { eval [list lappend res]\ [permute [lreplace \$list \$n \$n] [linsert \$prefix end \$e]] incr n } return \$res } [Vince] says that once upon a time he came across a very clever algorithm which had a state-based approach to generating permutations, which worked something like this: while {[getNextPermutation privateState permVar]} { puts \$permVar } i.e. the procedure maintains some state 'privateState' which allows it to iterate through the permutations one by one, and therefore avoids the need to pass in a ''\$script'' as in the foreach-permutation example above. Unfortunately, I can't remember that algorithm right now... [RS]: nice challenge! I don't claim to have the optimal variant, but the following code snippets work for me. If you normalize the set to be permuted to integers 0 < i < 10, you can write each permutation as a decimal number by just slapping the digits together. These decimals are ordered increasingly, with deltas that seem always to be multiples of 9. 1 2 3 4 -> 1234 1 2 4 3 -> 1243 ... First, a function to check whether an integer is a "perm" number, i.e. contains all the digits 1..n, where n is its number of digits: proc isperm x { foreach i [iota [string length \$x]] { if ![contains [incr i] \$x] {return 0} } return 1 } A little [integer range generator] - [[iota 4]] -> {0 1 2 3} proc iota n { set res {} for {set i 0} {\$i<\$n} {incr i} {lappend res \$i} set res } A tight wrapper for substring containment: proc contains {substr string} {expr {[string first \$substr \$string]>=0}} So given one permutation number, we add 9 as often as needed to reach the next perm number proc nextperm x { set length [string length \$x] while 1 { if [isperm [incr x 9]] {return \$x} if {[string length \$x]>\$length} return } } When at end (no further permutation possible), an empty string is returned. ---- [KBK] 2005-02-17 : In response to a request from [RS], here's a pair of procedures that return the lexicographically first permutation of a set of elements, and the lexicographically next permutation given the current permutation. The general principle is: * Empty lists and singletons have no "next permutation". * If you have a permutation {p q r s t}, the preferred choice for the "next permutation" is p, followed by the "next permutation" of {q r s t}. * If {q r s t} doesn't have a "next permutation", then the next possibility is to choose the element after p in sequence (without loss of generality, let it be q), and return it followed by the lexicographically first permutation of {p r s t}. * If p was the greatest element in the set {p q r s t}, and the above two tests fail, then we have the last permutation. # Procedure that forms the lexicographically first permutation of a list of # elements. proc firstperm { list } { lsort \$list } # Procedure that accepts a permutation of a set of elements and returns # the next permutatation in lexicographic sequence. The optional # "partial" arg is a list of elements that is prepended to the return # value. proc nextperm { perm { partial {}} } { # If a permutation is of a single element, there's no # "next permutation." if { [llength \$perm] <= 1 } { return {} } # Try to hold the first element fixed, and make the "next permutation" # of the remaining elements. set first [lindex \$perm 0] set p2 \$partial lappend p2 \$first set next [nextperm [lrange \$perm 1 end] \$p2] if {[llength \$next] > 0} { return \$next } # If the remaining elements were in descending sequence (that is, # were the last permutation of those elements), choose the # lexicographically next "first element". Fail if the "first element" # of the permutation was the lexicographically first. set elements [lsort \$perm] set idx [lsearch -exact \$elements \$first] incr idx if { \$idx >= [llength \$elements] } { return {} } # Place the new first element at the head of the permutation, and # follow with the remaining choices in ascending order. set ret \$partial lappend ret [lindex \$elements \$idx] foreach e [lreplace \$elements \$idx \$idx] { lappend ret \$e } return \$ret } # Demonstration - permute four elements for { set p [firstperm {alfa bravo charlie delta}] } \ { [llength \$p] > 0 } \ { set p [nextperm \$p] } \ { puts \$p } ---- [RS]: Ah, Kevin beat me to it... Here's my solution, based on the observation that reorganisation starts at the last pair of ascending neighbors - from that position, the minmal element greater than the smaller neighbor is moved to front, and the rest sorted: proc nextperm perm { #-- determine last ascending neighbors set last "" for {set i 0} {\$i<[llength \$perm]-1} {incr i} { if {[lindex \$perm \$i]<[lindex \$perm [expr {\$i+1}]]} { set last \$i } } if {\$last ne ""} { set pivot [lindex \$perm \$last] set successors [lrange \$perm \$last end] set minSucc [mingt \$successors \$pivot] concat [lrange \$perm 0 [expr {\$last-1}]] \$minSucc \ [lsort [l- \$successors \$minSucc]] } } This generally useful function removes an element from a list by value: proc l- {list element} { set pos [lsearch -exact \$list \$element] lreplace \$list \$pos \$pos } This searches ''list'' for the minimum element greater than ''pivot'': proc mingt {list pivot} { set res "" foreach i \$list { if {\$i>\$pivot && (\$res eq "" || \$i<\$res)} { set res \$i } } set res } The code passes both numeric and non-numeric tests, producing the ordered sequence of permutations like Kevin's test: for {set set [lsort {Tom Dick Harry Bob}]} {\$set ne ""} {} { puts \$set; set set [nextperm \$set] } for {set set [lsort {1 2 3 4}]} {\$set ne ""} {} { puts \$set; set set [nextperm \$set] } So let's compare the two versions! (I renamed them by author) proc try {cmd set} { for {set perm [lsort \$set]} {[llength \$perm]} {} {set perm [\$cmd \$perm]} } % time {try nextpermRS {1 2 3 4}} 100 21001 microseconds per iteration % time {try nextpermKBK {1 2 3 4}} 100 18305 microseconds per iteration After replacing two [info exists] tests with comparison against initial "", I however get % time {try nextpermRS {1 2 3 4}} 100 15155 microseconds per iteration ---- [Category Mathematics]