Synnefo » snf-ganeti

HBAL(1) Ganeti | Version @GANETI_VERSION@

=========================================

NAME

----

hbal \- Cluster balancer for Ganeti

SYNOPSIS

--------

**hbal** {backend options...} [algorithm options...] [reporting options...]

**hbal** --version

Backend options:

{ **-m** *cluster* | **-L[** *path* **] [-X]** | **-t** *data-file* }

Algorithm options:

**[ --max-cpu *cpu-ratio* ]**

**[ --min-disk *disk-ratio* ]**

**[ -l *limit* ]**

**[ -e *score* ]**

**[ -g *delta* ]** **[ --min-gain-limit *threshold* ]**

**[ -O *name...* ]**

**[ --no-disk-moves ]**

**[ --no-instance-moves ]**

**[ -U *util-file* ]**

**[ --evac-mode ]**

**[ --select-instances *inst...* ]**

**[ --exclude-instances *inst...* ]**

Reporting options:

**[ -C[ *file* ] ]**

**[ -p[ *fields* ] ]**

**[ --print-instances ]**

**[ -o ]**

**[ -v... | -q ]**

DESCRIPTION

-----------

hbal is a cluster balancer that looks at the current state of the

cluster (nodes with their total and free disk, memory, etc.) and

instance placement and computes a series of steps designed to bring

the cluster into a better state.

The algorithm used is designed to be stable (i.e. it will give you the

same results when restarting it from the middle of the solution) and

reasonably fast. It is not, however, designed to be a perfect

algorithm--it is possible to make it go into a corner from which

it can find no improvement, because it looks only one "step" ahead.

By default, the program will show the solution incrementally as it is

computed, in a somewhat cryptic format; for getting the actual Ganeti

command list, use the **-C** option.

ALGORITHM

~~~~~~~~~

The program works in independent steps; at each step, we compute the

best instance move that lowers the cluster score.

The possible move type for an instance are combinations of

failover/migrate and replace-disks such that we change one of the

instance nodes, and the other one remains (but possibly with changed

role, e.g. from primary it becomes secondary). The list is:

- failover (f)

- replace secondary (r)

- replace primary, a composite move (f, r, f)

- failover and replace secondary, also composite (f, r)

- replace secondary and failover, also composite (r, f)

We don't do the only remaining possibility of replacing both nodes

(r,f,r,f or the equivalent f,r,f,r) since these move needs an

exhaustive search over both candidate primary and secondary nodes, and

is O(n*n) in the number of nodes. Furthermore, it doesn't seems to

give better scores but will result in more disk replacements.

PLACEMENT RESTRICTIONS

~~~~~~~~~~~~~~~~~~~~~~

At each step, we prevent an instance move if it would cause:

- a node to go into N+1 failure state

- an instance to move onto an offline node (offline nodes are either

  read from the cluster or declared with *-O*)

- an exclusion-tag based conflict (exclusion tags are read from the

  cluster and/or defined via the *--exclusion-tags* option)

- a max vcpu/pcpu ratio to be exceeded (configured via *--max-cpu*)

- min disk free percentage to go below the configured limit

  (configured via *--min-disk*)

CLUSTER SCORING

~~~~~~~~~~~~~~~

As said before, the algorithm tries to minimise the cluster score at

each step. Currently this score is computed as a sum of the following

components:

- standard deviation of the percent of free memory

- standard deviation of the percent of reserved memory

- standard deviation of the percent of free disk

- count of nodes failing N+1 check

- count of instances living (either as primary or secondary) on

  offline nodes

- count of instances living (as primary) on offline nodes; this

  differs from the above metric by helping failover of such instances

  in 2-node clusters

- standard deviation of the ratio of virtual-to-physical cpus (for

  primary instances of the node)

- standard deviation of the dynamic load on the nodes, for cpus,

  memory, disk and network

The free memory and free disk values help ensure that all nodes are

somewhat balanced in their resource usage. The reserved memory helps

to ensure that nodes are somewhat balanced in holding secondary

instances, and that no node keeps too much memory reserved for

N+1. And finally, the N+1 percentage helps guide the algorithm towards

eliminating N+1 failures, if possible.

Except for the N+1 failures and offline instances counts, we use the

standard deviation since when used with values within a fixed range

(we use percents expressed as values between zero and one) it gives

consistent results across all metrics (there are some small issues

related to different means, but it works generally well). The 'count'

type values will have higher score and thus will matter more for

balancing; thus these are better for hard constraints (like evacuating

nodes and fixing N+1 failures). For example, the offline instances

count (i.e. the number of instances living on offline nodes) will

cause the algorithm to actively move instances away from offline

nodes. This, coupled with the restriction on placement given by

offline nodes, will cause evacuation of such nodes.

The dynamic load values need to be read from an external file (Ganeti

doesn't supply them), and are computed for each node as: sum of

primary instance cpu load, sum of primary instance memory load, sum of

primary and secondary instance disk load (as DRBD generates write load

on secondary nodes too in normal case and in degraded scenarios also

read load), and sum of primary instance network load. An example of

how to generate these values for input to hbal would be to track ``xm

list`` for instances over a day and by computing the delta of the cpu

values, and feed that via the *-U* option for all instances (and keep

the other metrics as one). For the algorithm to work, all that is

needed is that the values are consistent for a metric across all

instances (e.g. all instances use cpu% to report cpu usage, and not

something related to number of CPU seconds used if the CPUs are

different), and that they are normalised to between zero and one. Note

that it's recommended to not have zero as the load value for any

instance metric since then secondary instances are not well balanced.

On a perfectly balanced cluster (all nodes the same size, all

instances the same size and spread across the nodes equally), the

values for all metrics would be zero. This doesn't happen too often in

practice :)

OFFLINE INSTANCES

~~~~~~~~~~~~~~~~~

Since current Ganeti versions do not report the memory used by offline

(down) instances, ignoring the run status of instances will cause

wrong calculations. For this reason, the algorithm subtracts the

memory size of down instances from the free node memory of their

primary node, in effect simulating the startup of such instances.

EXCLUSION TAGS

~~~~~~~~~~~~~~

The exclusion tags mechanism is designed to prevent instances which

run the same workload (e.g. two DNS servers) to land on the same node,

which would make the respective node a SPOF for the given service.

It works by tagging instances with certain tags and then building

exclusion maps based on these. Which tags are actually used is

configured either via the command line (option *--exclusion-tags*)

or via adding them to the cluster tags:

--exclusion-tags=a,b

  This will make all instance tags of the form *a:\**, *b:\** be

  considered for the exclusion map

cluster tags *htools:iextags:a*, *htools:iextags:b*

  This will make instance tags *a:\**, *b:\** be considered for the

  exclusion map. More precisely, the suffix of cluster tags starting

  with *htools:iextags:* will become the prefix of the exclusion tags.

Both the above forms mean that two instances both having (e.g.) the

tag *a:foo* or *b:bar* won't end on the same node.

OPTIONS

-------

The options that can be passed to the program are as follows:

-C, --print-commands

  Print the command list at the end of the run. Without this, the

  program will only show a shorter, but cryptic output.

  Note that the moves list will be split into independent steps,

  called "jobsets", but only for visual inspection, not for actually

  parallelisation. It is not possible to parallelise these directly

  when executed via "gnt-instance" commands, since a compound command

  (e.g. failover and replace-disks) must be executed

  serially. Parallel execution is only possible when using the Luxi

  backend and the *-L* option.

  The algorithm for splitting the moves into jobsets is by

  accumulating moves until the next move is touching nodes already

  touched by the current moves; this means we can't execute in

  parallel (due to resource allocation in Ganeti) and thus we start a

  new jobset.

-p, --print-nodes

  Prints the before and after node status, in a format designed to

  allow the user to understand the node's most important parameters.

  It is possible to customise the listed information by passing a

  comma-separated list of field names to this option (the field list

  is currently undocumented), or to extend the default field list by

  prefixing the additional field list with a plus sign. By default,

  the node list will contain the following information:

  F

    a character denoting the status of the node, with '-' meaning an

    offline node, '*' meaning N+1 failure and blank meaning a good

    node

  Name

    the node name

  t_mem

    the total node memory

  n_mem

    the memory used by the node itself

  i_mem

    the memory used by instances

  x_mem

    amount memory which seems to be in use but cannot be determined

    why or by which instance; usually this means that the hypervisor

    has some overhead or that there are other reporting errors

  f_mem

    the free node memory

  r_mem

    the reserved node memory, which is the amount of free memory

    needed for N+1 compliance

  t_dsk

    total disk

  f_dsk

    free disk

  pcpu

    the number of physical cpus on the node

  vcpu

    the number of virtual cpus allocated to primary instances

  pcnt

    number of primary instances

  scnt

    number of secondary instances

  p_fmem

    percent of free memory

  p_fdsk

    percent of free disk

  r_cpu

    ratio of virtual to physical cpus

  lCpu

    the dynamic CPU load (if the information is available)

  lMem

    the dynamic memory load (if the information is available)

  lDsk

    the dynamic disk load (if the information is available)

  lNet

    the dynamic net load (if the information is available)

--print-instances

  Prints the before and after instance map. This is less useful as the

  node status, but it can help in understanding instance moves.

-o, --oneline

  Only shows a one-line output from the program, designed for the case

  when one wants to look at multiple clusters at once and check their

  status.

  The line will contain four fields:

  - initial cluster score

  - number of steps in the solution

  - final cluster score

  - improvement in the cluster score

-O *name*

  This option (which can be given multiple times) will mark nodes as

  being *offline*. This means a couple of things:

  - instances won't be placed on these nodes, not even temporarily;

    e.g. the *replace primary* move is not available if the secondary

    node is offline, since this move requires a failover.

  - these nodes will not be included in the score calculation (except

    for the percentage of instances on offline nodes)

  Note that algorithm will also mark as offline any nodes which are

  reported by RAPI as such, or that have "?" in file-based input in

  any numeric fields.

-e *score*, --min-score=*score*

  This parameter denotes the minimum score we are happy with and alters

  the computation in two ways:

  - if the cluster has the initial score lower than this value, then we

    don't enter the algorithm at all, and exit with success

  - during the iterative process, if we reach a score lower than this

    value, we exit the algorithm

  The default value of the parameter is currently ``1e-9`` (chosen

  empirically).

-g *delta*, --min-gain=*delta*

  Since the balancing algorithm can sometimes result in just very tiny

  improvements, that bring less gain that they cost in relocation

  time, this parameter (defaulting to 0.01) represents the minimum

  gain we require during a step, to continue balancing.

--min-gain-limit=*threshold*

  The above min-gain option will only take effect if the cluster score

  is already below *threshold* (defaults to 0.1). The rationale behind

  this setting is that at high cluster scores (badly balanced

  clusters), we don't want to abort the rebalance too quickly, as

  later gains might still be significant. However, under the

  threshold, the total gain is only the threshold value, so we can

  exit early.

--no-disk-moves

  This parameter prevents hbal from using disk move

  (i.e. "gnt-instance replace-disks") operations. This will result in

  a much quicker balancing, but of course the improvements are

  limited. It is up to the user to decide when to use one or another.

--no-instance-moves

  This parameter prevents hbal from using instance moves

  (i.e. "gnt-instance migrate/failover") operations. This will only use

  the slow disk-replacement operations, and will also provide a worse

  balance, but can be useful if moving instances around is deemed unsafe

  or not preferred.

--evac-mode

  This parameter restricts the list of instances considered for moving

  to the ones living on offline/drained nodes. It can be used as a

  (bulk) replacement for Ganeti's own *gnt-node evacuate*, with the

  note that it doesn't guarantee full evacuation.

--select-instances=*instances*

  This parameter marks the given instances (as a comma-separated list)

  as the only ones being moved during the rebalance.

--exclude-instances=*instances*

  This parameter marks the given instances (as a comma-separated list)

  from being moved during the rebalance.

-U *util-file*

  This parameter specifies a file holding instance dynamic utilisation

  information that will be used to tweak the balancing algorithm to

  equalise load on the nodes (as opposed to static resource

  usage). The file is in the format "instance_name cpu_util mem_util

  disk_util net_util" where the "_util" parameters are interpreted as

  numbers and the instance name must match exactly the instance as

  read from Ganeti. In case of unknown instance names, the program

  will abort.

  If not given, the default values are one for all metrics and thus

  dynamic utilisation has only one effect on the algorithm: the

  equalisation of the secondary instances across nodes (this is the

  only metric that is not tracked by another, dedicated value, and

  thus the disk load of instances will cause secondary instance

  equalisation). Note that value of one will also influence slightly

  the primary instance count, but that is already tracked via other

  metrics and thus the influence of the dynamic utilisation will be

  practically insignificant.

-t *datafile*, --text-data=*datafile*

  The name of the file holding node and instance information (if not

  collecting via RAPI or LUXI). This or one of the other backends must

  be selected.

-S *filename*, --save-cluster=*filename*

  If given, the state of the cluster before the balancing is saved to

  the given file plus the extension "original"

  (i.e. *filename*.original), and the state at the end of the

  balancing is saved to the given file plus the extension "balanced"

  (i.e. *filename*.balanced). This allows re-feeding the cluster state

  to either hbal itself or for example hspace.

-m *cluster*

 Collect data directly from the *cluster* given as an argument via

 RAPI. If the argument doesn't contain a colon (:), then it is

 converted into a fully-built URL via prepending ``https://`` and

 appending the default RAPI port, otherwise it's considered a

 fully-specified URL and is used as-is.

-L [*path*]

  Collect data directly from the master daemon, which is to be

  contacted via the luxi (an internal Ganeti protocol). An optional

  *path* argument is interpreted as the path to the unix socket on

  which the master daemon listens; otherwise, the default path used by

  ganeti when installed with *--localstatedir=/var* is used.

-X

  When using the Luxi backend, hbal can also execute the given

  commands. The execution method is to execute the individual jobsets

  (see the *-C* option for details) in separate stages, aborting if at

  any time a jobset doesn't have all jobs successful. Each step in the

  balancing solution will be translated into exactly one Ganeti job

  (having between one and three OpCodes), and all the steps in a

  jobset will be executed in parallel. The jobsets themselves are

  executed serially.

-l *N*, --max-length=*N*

  Restrict the solution to this length. This can be used for example

  to automate the execution of the balancing.

--max-cpu=*cpu-ratio*

  The maximum virtual to physical cpu ratio, as a floating point number

  greater than or equal to one. For example, specifying *cpu-ratio* as

  **2.5** means that, for a 4-cpu machine, a maximum of 10 virtual cpus

  should be allowed to be in use for primary instances. A value of

  exactly one means there will be no over-subscription of CPU (except

  for the CPU time used by the node itself), and values below one do not

  make sense, as that means other resources (e.g. disk) won't be fully

  utilised due to CPU restrictions.

--min-disk=*disk-ratio*

  The minimum amount of free disk space remaining, as a floating point

  number. For example, specifying *disk-ratio* as **0.25** means that

  at least one quarter of disk space should be left free on nodes.

-G *uuid*, --group=*uuid*

  On an multi-group cluster, select this group for

  processing. Otherwise hbal will abort, since it cannot balance

  multiple groups at the same time.

-v, --verbose

  Increase the output verbosity. Each usage of this option will

  increase the verbosity (currently more than 2 doesn't make sense)

  from the default of one.

-q, --quiet

  Decrease the output verbosity. Each usage of this option will

  decrease the verbosity (less than zero doesn't make sense) from the

  default of one.

-V, --version

  Just show the program version and exit.

EXIT STATUS

-----------

The exit status of the command will be zero, unless for some reason

the algorithm fatally failed (e.g. wrong node or instance data), or

(in case of job execution) any job has failed.

BUGS

----

The program does not check its input data for consistency, and aborts

with cryptic errors messages in this case.

The algorithm is not perfect.

The output format is not easily scriptable, and the program should

feed moves directly into Ganeti (either via RAPI or via a gnt-debug

input file).

EXAMPLE

-------

Note that these examples are not for the latest version (they don't

have full node data).

Default output

~~~~~~~~~~~~~~

With the default options, the program shows each individual step and

the improvements it brings in cluster score::

    $ hbal

    Loaded 20 nodes, 80 instances

    Cluster is not N+1 happy, continuing but no guarantee that the cluster will end N+1 happy.

    Initial score: 0.52329131

    Trying to minimize the CV...

        1. instance14  node1:node10  => node16:node10 0.42109120 a=f r:node16 f

        2. instance54  node4:node15  => node16:node15 0.31904594 a=f r:node16 f

        3. instance4   node5:node2   => node2:node16  0.26611015 a=f r:node16

        4. instance48  node18:node20 => node2:node18  0.21361717 a=r:node2 f

        5. instance93  node19:node18 => node16:node19 0.16166425 a=r:node16 f

        6. instance89  node3:node20  => node2:node3   0.11005629 a=r:node2 f

        7. instance5   node6:node2   => node16:node6  0.05841589 a=r:node16 f

        8. instance94  node7:node20  => node20:node16 0.00658759 a=f r:node16

        9. instance44  node20:node2  => node2:node15  0.00438740 a=f r:node15

       10. instance62  node14:node18 => node14:node16 0.00390087 a=r:node16

       11. instance13  node11:node14 => node11:node16 0.00361787 a=r:node16

       12. instance19  node10:node11 => node10:node7  0.00336636 a=r:node7

       13. instance43  node12:node13 => node12:node1  0.00305681 a=r:node1

       14. instance1   node1:node2   => node1:node4   0.00263124 a=r:node4

       15. instance58  node19:node20 => node19:node17 0.00252594 a=r:node17

    Cluster score improved from 0.52329131 to 0.00252594

In the above output, we can see:

- the input data (here from files) shows a cluster with 20 nodes and

  80 instances

- the cluster is not initially N+1 compliant

- the initial score is 0.52329131

The step list follows, showing the instance, its initial

primary/secondary nodes, the new primary secondary, the cluster list,

and the actions taken in this step (with 'f' denoting failover/migrate

and 'r' denoting replace secondary).

Finally, the program shows the improvement in cluster score.

A more detailed output is obtained via the *-C* and *-p* options::

    $ hbal

    Loaded 20 nodes, 80 instances

    Cluster is not N+1 happy, continuing but no guarantee that the cluster will end N+1 happy.

    Initial cluster status:

    N1 Name   t_mem f_mem r_mem t_dsk f_dsk pri sec  p_fmem  p_fdsk

     * node1  32762  1280  6000  1861  1026   5   3 0.03907 0.55179

       node2  32762 31280 12000  1861  1026   0   8 0.95476 0.55179

     * node3  32762  1280  6000  1861  1026   5   3 0.03907 0.55179

     * node4  32762  1280  6000  1861  1026   5   3 0.03907 0.55179

     * node5  32762  1280  6000  1861   978   5   5 0.03907 0.52573

     * node6  32762  1280  6000  1861  1026   5   3 0.03907 0.55179

     * node7  32762  1280  6000  1861  1026   5   3 0.03907 0.55179

       node8  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node9  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

     * node10 32762  7280 12000  1861  1026   4   4 0.22221 0.55179

       node11 32762  7280  6000  1861   922   4   5 0.22221 0.49577

       node12 32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node13 32762  7280  6000  1861   922   4   5 0.22221 0.49577

       node14 32762  7280  6000  1861   922   4   5 0.22221 0.49577

     * node15 32762  7280 12000  1861  1131   4   3 0.22221 0.60782

       node16 32762 31280     0  1861  1860   0   0 0.95476 1.00000

       node17 32762  7280  6000  1861  1106   5   3 0.22221 0.59479

     * node18 32762  1280  6000  1396   561   5   3 0.03907 0.40239

     * node19 32762  1280  6000  1861  1026   5   3 0.03907 0.55179

       node20 32762 13280 12000  1861   689   3   9 0.40535 0.37068

    Initial score: 0.52329131

    Trying to minimize the CV...

        1. instance14  node1:node10  => node16:node10 0.42109120 a=f r:node16 f

        2. instance54  node4:node15  => node16:node15 0.31904594 a=f r:node16 f

        3. instance4   node5:node2   => node2:node16  0.26611015 a=f r:node16

        4. instance48  node18:node20 => node2:node18  0.21361717 a=r:node2 f

        5. instance93  node19:node18 => node16:node19 0.16166425 a=r:node16 f

        6. instance89  node3:node20  => node2:node3   0.11005629 a=r:node2 f

        7. instance5   node6:node2   => node16:node6  0.05841589 a=r:node16 f

        8. instance94  node7:node20  => node20:node16 0.00658759 a=f r:node16

        9. instance44  node20:node2  => node2:node15  0.00438740 a=f r:node15

       10. instance62  node14:node18 => node14:node16 0.00390087 a=r:node16

       11. instance13  node11:node14 => node11:node16 0.00361787 a=r:node16

       12. instance19  node10:node11 => node10:node7  0.00336636 a=r:node7

       13. instance43  node12:node13 => node12:node1  0.00305681 a=r:node1

       14. instance1   node1:node2   => node1:node4   0.00263124 a=r:node4

       15. instance58  node19:node20 => node19:node17 0.00252594 a=r:node17

    Cluster score improved from 0.52329131 to 0.00252594

    Commands to run to reach the above solution:

      echo step 1

      echo gnt-instance migrate instance14

      echo gnt-instance replace-disks -n node16 instance14

      echo gnt-instance migrate instance14

      echo step 2

      echo gnt-instance migrate instance54

      echo gnt-instance replace-disks -n node16 instance54

      echo gnt-instance migrate instance54

      echo step 3

      echo gnt-instance migrate instance4

      echo gnt-instance replace-disks -n node16 instance4

      echo step 4

      echo gnt-instance replace-disks -n node2 instance48

      echo gnt-instance migrate instance48

      echo step 5

      echo gnt-instance replace-disks -n node16 instance93

      echo gnt-instance migrate instance93

      echo step 6

      echo gnt-instance replace-disks -n node2 instance89

      echo gnt-instance migrate instance89

      echo step 7

      echo gnt-instance replace-disks -n node16 instance5

      echo gnt-instance migrate instance5

      echo step 8

      echo gnt-instance migrate instance94

      echo gnt-instance replace-disks -n node16 instance94

      echo step 9

      echo gnt-instance migrate instance44

      echo gnt-instance replace-disks -n node15 instance44

      echo step 10

      echo gnt-instance replace-disks -n node16 instance62

      echo step 11

      echo gnt-instance replace-disks -n node16 instance13

      echo step 12

      echo gnt-instance replace-disks -n node7 instance19

      echo step 13

      echo gnt-instance replace-disks -n node1 instance43

      echo step 14

      echo gnt-instance replace-disks -n node4 instance1

      echo step 15

      echo gnt-instance replace-disks -n node17 instance58

    Final cluster status:

    N1 Name   t_mem f_mem r_mem t_dsk f_dsk pri sec  p_fmem  p_fdsk

       node1  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node2  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node3  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node4  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node5  32762  7280  6000  1861  1078   4   5 0.22221 0.57947

       node6  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node7  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node8  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node9  32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node10 32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node11 32762  7280  6000  1861  1022   4   4 0.22221 0.54951

       node12 32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node13 32762  7280  6000  1861  1022   4   4 0.22221 0.54951

       node14 32762  7280  6000  1861  1022   4   4 0.22221 0.54951

       node15 32762  7280  6000  1861  1031   4   4 0.22221 0.55408

       node16 32762  7280  6000  1861  1060   4   4 0.22221 0.57007

       node17 32762  7280  6000  1861  1006   5   4 0.22221 0.54105

       node18 32762  7280  6000  1396   761   4   2 0.22221 0.54570

       node19 32762  7280  6000  1861  1026   4   4 0.22221 0.55179

       node20 32762 13280  6000  1861  1089   3   5 0.40535 0.58565

Here we see, beside the step list, the initial and final cluster

status, with the final one showing all nodes being N+1 compliant, and

the command list to reach the final solution. In the initial listing,

we see which nodes are not N+1 compliant.

The algorithm is stable as long as each step above is fully completed,

e.g. in step 8, both the migrate and the replace-disks are

done. Otherwise, if only the migrate is done, the input data is

changed in a way that the program will output a different solution

list (but hopefully will end in the same state).

.. vim: set textwidth=72 :

.. Local Variables:

.. mode: rst

.. fill-column: 72

.. End: