The Algorithm
The search strategy used by PENNY starts by making a tree consisting of the first two species (the first three if the tree is to be unrooted). Then it tries to add the next species in all possible places (there are three of these). For each of the resulting trees it evaluates the number of steps. It adds the next species to each of these, again in all possible spaces. If this process would continue it would simply generate all possible trees, of which there are a very large number even when the number of species is moderate (34,459,425 with 10 species). Actually it does not do this, because the trees are generated in a particular order and some of them are never generated.
Actually the order in which trees are generated is not quite as implied above, but is a "depth-first search". This means that first one adds the third species in the first possible place, then the fourth species in its first possible place, then the fifth and so on until the first possible tree has been produced. Its number of steps is evaluated. Then one "backtracks" by trying the alternative placements of the last species. When these are exhausted one tries the next placement of the next-to-last species. The order of placement in a depth-first search is like this for a four-species case (parentheses enclose monophyletic groups):
Make tree of first two species: (A,B)
Add C in first place: ((A,B),C)
Add D in first place: (((A,D),B),C)
Add D in second place: ((A,(B,D)),C)
Add D in third place: (((A,B),D),C)
Add D in fourth place: ((A,B),(C,D))
Add D in fifth place: (((A,B),C),D)
Add C in second place: ((A,C),B)
Add D in first place: (((A,D),C),B)
Add D in second place: ((A,(C,D)),B)
Add D in third place: (((A,C),D),B)
Add D in fourth place: ((A,C),(B,D))
Add D in fifth place: (((A,C),B),D)
Add C in third place: (A,(B,C))
Add D in first place: ((A,D),(B,C))
Add D in second place: (A,((B,D),C))
Add D in third place: (A,(B,(C,D)))
Add D in fourth place: (A,((B,C),D))
Add D in fifth place: ((A,(B,C)),D)
Among these fifteen trees you will find all of the four-species rooted
bifurcating trees, each exactly once (the parentheses each enclose a
monophyletic group). As displayed above, the backtracking depth-first search
algorithm is just another way of producing all possible trees one at a time.
The branch and bound algorithm consists of this with one change. As each tree
is constructed, including the partial trees such as (A,(B,C)), its number of
steps is evaluated. In addition a prediction is made as to how many steps will
be added, at a minimum, as further species are added.
This is done by counting how many binary characters which are invariant in the data up the species most recently added will ultimately show variation when further species are added. Thus if 20 characters vary among species A, B, and C and their root, and if tree ((A,C),B) requires 24 steps, then if there are 8 more characters which will be seen to vary when species D is added, we can immediately say that no matter how we add D, the resulting tree can have no less than 24 + 8 = 32 steps. The point of all this is that if a previously- found tree such as ((A,B),(C,D)) required only 30 steps, then we know that there is no point in even trying to add D to ((A,C),B). We have computed the bound that enables us to cut off a whole line of inquiry (in this case five trees) and avoid going down that particular branch any farther.
The branch-and-bound algorithm thus allows us to find all most parsimonious trees without generating all possible trees. How much of a saving this is depends strongly on the data. For very clean (nearly "Hennigian") data, it saves much time, but on very messy data it will still take a very long time.
The algorithm in the program differs from the one outlined here in some essential details: it investigates possibilities in the order of their apparent promise. This applies to the order of addition of species, and to the places where they are added to the tree. After the first two-species tree is constructed, the program tries adding each of the remaining species in turn, each in the best possible place it can find. Whichever of those species adds (at a minimum) the most additional steps is taken to be the one to be added next to the tree. When it is added, it is added in turn to places which cause the fewest additional steps to be added. This sounds a bit complex, but it is done with the intention of eliminating regions of the search of all possible trees as soon as possible, and lowering the bound on tree length as quickly as possible.
The program keeps a list of all the most parsimonious trees found so far. Whenever it finds one that has fewer steps than these, it clears out the list and restarts the list with that tree. In the process the bound tightens and fewer possibilities need be investigated. At the end the list contains all the shortest trees. These are then printed out. It should be mentioned that the program CLIQUE for finding all largest cliques also works by branch-and-bound. Both problems are NP-complete but for some reason CLIQUE runs far faster. Although their worst-case behavior is bad for both programs, those worst cases occur far more frequently in parsimony problems than in compatibility problems.
Among the quantities available to be set at the beginning of a run of
PENNY, two (howoften and howmany) are of particular importance. As PENNY goes
along it will keep count of how many trees it has examined. Suppose that
howoften is 100 and howmany is 1000, the default settings. Every time 100
trees have been examined, PENNY will print out a line saying how many multiples
of 100 trees have now been examined, how many steps the most parsimonious tree
found so far has, how many trees of with that number of steps have been found,
and a very rough estimate of what fraction of all trees have been looked at so
far.
When the number of these multiples printed out reaches the number howmany
(say 1000), the whole algorithm aborts and prints out that it has not found all
most parsimonious trees, but prints out what is has got so far anyway. These
trees need not be any of the most parsimonious trees: they are simply the most
parsimonious ones found so far. By setting the product (howoften X howmany)
large you can make the algorithm less likely to abort, but then you risk
getting bogged down in a gigantic computation. You should adjust these
constants so that the program cannot go beyond examining the number of trees
you are reasonably willing to pay for (or wait for). In their initial setting
the program will abort after looking at 100,000 trees. Obviously you may want
to adjust howoften in order to get more or fewer lines of intermediate notice
of how many trees have been looked at so far. Of course, in small cases you
may never even reach the first multiple of howoften and nothing will be printed
out except some headings and then the final trees.
The indication of the approximate percentage of trees searched so far will
be helpful in judging how much farther you would have to go to get the full
search. Actually, since that fraction is the fraction of the set of all
possible trees searched or ruled out so far, and since the search becomes
progressively more efficient, the approximate fraction printed out will usually
be an underestimate of how far along the program is, sometimes a serious
underestimate.
A constant that affects the result is "maxtrees", which controls the
maximum number of trees that can be stored. Thus if "maxtrees" is 25, and 32
most parsimonious trees are found, only the first 25 of these are stored and
printed out. If "maxtrees" is increased, the program does not run any slower
but requires a little more intermediate storage space. I recommend that
"maxtrees" be kept as large as you can, provided you are willing to look at an
output with that many trees on it! Initially, "maxtrees" is set to 100 in the
distribution copy.
The counting of the length of trees is done by an algorithm nearly
identical to the corresponding algorithms in MIX, and thus the remainder of
this document will be nearly identical to the MIX document. MIX is a general
parsimony program which carries out the Wagner and Camin-Sokal parsimony
methods in mixture, where each character can have its method specified. The
program defaults to carrying out Wagner parsimony.
The Camin-Sokal parsimony method explains the data by assuming that
changes 0 --> 1 are allowed but not changes 1 --> 0. Wagner parsimony allows
both kinds of changes. (This under the assumption that 0 is the ancestral
state, though the program allows reassignment of the ancestral state, in which
case we must reverse the state numbers 0 and 1 throughout this discussion).
The criterion is to find the tree which requires the minimum number of changes.
The Camin-Sokal method is due to
Camin and Sokal (1965) and the Wagner method
to
Eck and Dayhoff (1966) and to
Kluge and Farris (1969).
Here are the assumptions of these two methods:
That these are the assumptions of parsimony methods has been documented in a
series of papers of mine: (
1973a,
1978b,
1979,
1981b,
1983b,
1988b). For an
opposing view arguing that the parsimony methods make no substantive
assumptions such as these, see the papers by
Farris (1983) and
Sober (1983a,
1983b, 1988), but also read the exchange between
Felsenstein and Sober (1986).
The input for PENNY is the standard input for discrete characters
programs, described above in the
discrete characters programs documentation file.
States "?", "P", and "B" are allowed.
Most of the options are selected using a menu:
The option P toggles between the Camin-Sokal parsimony criterion and the
Wagner parsimony criterion. Options F and H reset the variables howoften (F)
and howmany (H). The user is prompted for the new values. By setting these
larger the program will report its progress less often (howoften) and will run
longer (howmany x often). These values default to 100 and 1000 which
guarantees a search of 100,000 trees, but these can be changed. Note that
option F in this program is not the Factors option available in some of the
other programs in this section of the package.
The A (Ancestral states) option works in the usual way, described in the
Discrete Characters Programs documentation file, and requires one (or more)
lines of auxiliary information specifying the states of the ancestor, as
described in those documentation files. It also requires that an A be present
on the first line of the input file. The ancestor is not to be counted as one
of the species. If the A option is not used, then the program will assume 0 as
the ancestral state for those characters following the Camin-Sokal method, and
will assume that the ancestral state is unknown for those characters following
Wagner parsimony. If any characters have unknown ancestral states, and if the
resulting tree is rooted (even by outgroup), a table will be printed out
showing the best guesses of which are the ancestral states in each character.
The S (Simple) option alters a step in PENNY which reconsiders the order
in which species are added to the tree. Normally the decision as to what
species to add to the tree next is made as the first tree is being
constructucted; that ordering of species is not altered subsequently. The S
option causes it to be continually reconsidered. This will probably result in
a substantial increase in run time, but on some data sets of intermediate
messiness it may help. It is included in case it might prove of use on some
data sets.
In the input file the W (Weights) option is available, as usual. The A
(Ancestral states) and X (Mixed methods) also require information to be present
and options declared on the first line of the input file. If the Ancestral
States option is invoked then you must also choose A from the menu. Note that
the X option requires that the option M (Mixture) be declared and the mixture
information provided in the input file. This is admittedly confusing. The F
(Factors) option is not available in this program, as it would have no effect
on the result even if that information were provided in the input file.
The final output is standard: a set of trees, which will be printed as
rooted or unrooted depending on which is appropriate, and if the user elects to
see them, tables of the number of changes of state required in each character.
If the Wagner option is in force for a character, it may not be possible to
unambiguously locate the places on the tree where the changes occur, as there
may be multiple possibilities. A table is available to be printed out after
each tree, showing for each branch whether there are known to be changes in the
branch, and what the states are inferred to have been at the top end of the
branch. If the inferred state is a "?" there will be multiple equally-
parsimonious assignments of states; the user must work these out for themselves
by hand.
If the Camin-Sokal parsimony method (option C or S) is invoked and the A
option is also used, then the program will infer, for any character whose
ancestral state is unknown ("?") whether the ancestral state 0 or 1 will give
the fewest state changes. If these are tied, then it may not be possible for
the program to infer the state in the internal nodes, and these will all be
printed as ".". If this has happened and you want to know more about the
states at the internal nodes, you will find helpful to use MOVE to display the
tree and examine its interior states, as the algorithm in MOVE shows all that
can be known in this case about the interior states, including where there is
and is not amibiguity. The algorithm in PENNY gives up more easily on
displaying these states.
If the A option is not used, then the program will assume 0 as the
ancestral state for those characters following the Camin-Sokal method, and will
assume that the ancestral state is unknown for those characters following
Wagner parsimony. If any characters have unknown ancestral states, and if the
resulting tree is rooted (even by outgroup), a table will be printed out
showing the best guesses of which are the ancestral states in each character.
You will find it useful to understand the difference between the Camin-Sokal
parsimony criterion with unknown ancestral state and the Wagner parsimony
criterion.
At the beginning of the program are a series of constants, which can be
changed to help adapt the program to different computer systems. Two are the
initial values of howmany and howoften, constants "often" and "many". Constant
"maxtrees" is the maximum number of tied trees that will be stored and
"nmlngth" is the length of the species names.
Controlling Run Times
Methods and Options
Penny algorithm, version 3.5c
branch-and-bound to find all most parsimonious trees
Settings for this run:
X Use Mixed method? No
P Parsimony method? Wagner
F How often to report, in trees: 100
H How many groups of 100 trees: 1000
O Outgroup root? No, use as outgroup species 1
S Branch and bound is simple? Yes
T Use Threshold parsimony? No, use ordinary parsimony
A Use ancestral states in input file? No
M Analyze multiple data sets? No
0 Terminal type (IBM PC, VT52, ANSI)? ANSI
1 Print out the data at start of run No
2 Print indications of progress of run Yes
3 Print out tree Yes
4 Print out steps in each character No
5 Print states at all nodes of tree No
6 Write out trees onto tree file? Yes
Are these settings correct? (type Y or the letter for one to change)
The options X, O, T, A, and M are the usual Mixed Methods, Outgroup,
Threshold, Ancestral States, and Multiple Data Sets options. They are
described in the Main documentation file and in the
Discrete Characters Programs documentation file.
The O option is only acted upon if the final tree
is unrooted.
TEST DATA SET
7 6
Alpha1 110110
Alpha2 110110
Beta1 110000
Beta2 110000
Gamma1 100110
Delta 001001
Epsilon 001110
TEST SET OUTPUT (with all numerical options turned on)
Penny algorithm, version 3.5c
branch-and-bound to find all most parsimonious trees
Wagner parsimony method
Name Characters
---- ----------
Alpha1 11011 0
Alpha2 11011 0
Beta1 11000 0
Beta2 11000 0
Gamma1 10011 0
Delta 00100 1
Epsilon 00111 0
requires a total of 8.000
3 trees in all found
+-----------------Alpha1
!
! +--------Alpha2
--1 !
! +-----4 +--Epsilon
! ! ! +--6
! ! +--5 +--Delta
+--2 !
! +-----Gamma1
!
! +--Beta2
+-----------3
+--Beta1
remember: this is an unrooted tree!
steps in each character:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 11011 0
1 Alpha1 no ..... .
1 2 no ..... .
2 4 no ..... .
4 Alpha2 no ..... .
4 5 yes .0... .
5 6 yes 0.1.. .
6 Epsilon no ..... .
6 Delta yes ...00 1
5 Gamma1 no ..... .
2 3 yes ...00 .
3 Beta2 no ..... .
3 Beta1 no ..... .
+-----------------Alpha1
!
--1 +--------------Alpha2
! !
! ! +--Epsilon
+--2 +--6
! +-----5 +--Delta
! ! !
+--4 +-----Gamma1
!
! +--Beta2
+--------3
+--Beta1
remember: this is an unrooted tree!
steps in each character:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 11011 0
1 Alpha1 no ..... .
1 2 no ..... .
2 Alpha2 no ..... .
2 4 no ..... .
4 5 yes .0... .
5 6 yes 0.1.. .
6 Epsilon no ..... .
6 Delta yes ...00 1
5 Gamma1 no ..... .
4 3 yes ...00 .
3 Beta2 no ..... .
3 Beta1 no ..... .
+-----------------Alpha1
!
! +-----Alpha2
--1 +--------2
! ! ! +--Beta2
! ! +--3
+--4 +--Beta1
!
! +--Epsilon
! +--6
+--------5 +--Delta
!
+-----Gamma1
remember: this is an unrooted tree!
steps in each character:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 11011 0
1 Alpha1 no ..... .
1 4 no ..... .
4 2 no ..... .
2 Alpha2 no ..... .
2 3 yes ...00 .
3 Beta2 no ..... .
3 Beta1 no ..... .
4 5 yes .0... .
5 6 yes 0.1.. .
6 Epsilon no ..... .
6 Delta yes ...00 1
5 Gamma1 no ..... .
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